Willeke Van Der Jagt-Jelsma

Margreet De Vries-Schot

Rint De Jong

Frank C Verhulst

Johan Ormel

René Veenstra

Sophie Swinkels

Jan Buitelaar

Willeke Van Der

The relationship between parental religiosity and mental health of pre-adolescents in a community sample. The TRAILS study

Keywords: mental health, pre-adolescents, religion, parents, harmony ABBREVATIONS: GLM = Generalized Linear Model, TRAILS = TRAcking Adolescents' Individual Lives Survey, CBCL = Child Behavior Checklist, YSR = Youth Self Report, TCP

Objective. The purpose of this study is to examine the relationship between parental religiosity, parental harmony on the subject of religiosity, and the mental health of pre-adolescents.

Methods. In a community-based sample of 2230 pre-adolescents (10-12 years) mental health problems were assessed using self-report (Youth Self-Report, YSR), parental report (Child Behavior Checklist, CBCL) as well as teacher report (Teacher Checklist for Psychopathology, TCP). Information about the religiosity of mother, the religiosity of father and religious harmony between the parents was obtained by parent report. Results. The influence of maternal religiosity on internalizing symptoms depended on the religious harmony between parents. This was particularly apparent on the CBCL. Higher levels of internalizing symptoms were associated with parental religious disharmony when combined with passive maternal religiosity. Boys scored themselves as having more externalizing symptoms in case of religiously disharmonious parents. The levels of internalizing and externalizing symptoms in pre-adolescents were not influenced by parental religiosity.

Conclusions. Religious disharmony between parents is a risk factor for internalizing problems when the mother is passive religious. Religious disharmony is a risk factor on its own for externalizing problems amongst boys. Parental religious activity and parental harmony play a role in the mental health of pre-adolescents.

Introduction

Religiosity is an important aspect of many individuals' lives, providing answers to fundamental questions relating to the meaning of life. By shaping fundamental beliefs and behaviors, religiosity may influence the mental health of individuals. The 4 th edition of the Diagnostic and Statistic Manual of Mental Disorders in 1994 introduced a new category because of this, namely V62.89 'religious or spiritual problem' [START_REF]Diagnostic and statistical manual of mental disorders[END_REF]. Throughout the child's development, the religiosity of the parents is also of importance. Parental religiosity and religious identity serve as important early resources upon which children draw when forming their own religious identity, developing their religious beliefs, and shaping their behaviors.

Few studies have examined the relationship between religiosity and mental health in children prior to adolescence. In a study among more than 2,500 11-year-old pre-adolescents from the two main Christian denominations in West Scotland, the relationship between church attendance and mental health was analyzed [START_REF] Abbotts | Is going to church good or bad for you? Denomination, attendance and mental health of children in West Scotland[END_REF]. Mental health questions were answered by children, parents and teachers. Levels of church-attendance were low among those affiliated with the Church of Scotland and relatively high among Catholics. Aggression was found to be less prevalent among weekly attendees of both denominations. Weekly church attendance was associated with better self-esteem, less anxiety and less depression for Catholics and with less self-esteem, more anxiety and more depression for children with a Church of Scotland affiliation. The only mental health measure to show a similar relationship with church attendance in both denominations was aggression, which was less prevalent among weekly attendees. Also, aggression was consistently scored by children, parents and teachers. However, the results on depression showed a lack of parental awareness of depression in their children. It is suggested that the key issue for these results could be whether church attendance is normative within the peer group, as it is for Catholics in this context. Over three-quarters of the studies on the relationship between religion and mental health of adolescents and adults report modest positive effects of religion [START_REF] Varon | Relationship between maternal church attendance and adolescent mental health and social functioning[END_REF][START_REF] Wong | A systematic review of recent research on adolescent religiosity/spirituality and mental health[END_REF][START_REF] Rew | The relationship between prayer, health behaviors, and protective resources in school-age children[END_REF][START_REF] Cotton | Religion/spirituality and adolescent health outcomes: a review[END_REF][START_REF] Hackney | Religiosity and Mental Health: A Meta-Analysis of Recent Studies[END_REF]. In contrast, a minority of studies report a negative relationship between religion and mental health. These conflicting results might have been caused by differences in populations studied or differences in defining and measuring religiosity and mental health. No longitudinal studies on the relationship between religion and mental health have been found, which implicates that causality as well as selection could be the underlying mechanism for the associations found between religion and mental health.

Greater parental religiosity also influences general indices of family functioning, and has been related to more cohesive family relationships, greater parenting skills, lower inter-parental conflicts, and fewer externalizing and internalizing problems among adolescents [START_REF] Brody | Parental religiosity, family processes, and youth competence in rural, two-parent African American families[END_REF]. Another study suggests that religiosity facilitates family cohesiveness when it is shared among family members.

However, if religiosity is not shared, this disharmony may result in higher levels of adolescent delinquent behaviors [START_REF] Pearce | Intergenerational religious dynamics and adolescent delinquency[END_REF]. On its own, family discord is a risk factor for various mental health problems like major depressive disorder, substance abuse and conduct disorders [START_REF] Pilowsky | Family discord, parental depression, and psychopathology in offspring: 20-year follow up[END_REF][START_REF] Meyer | Familial aggregation for conduct disorder symptomatology: the role of genes, marital discord and family adaptability[END_REF][START_REF] Rutter | Family discord and conduct disorder: cause, consequence, or correlate?[END_REF]. Research involving adolescents indicates that parental religiosity protects girls considerably better than boys against delinquency. The protective effect concerning delinquency tends to diminish as the discord between parents regarding religion increases [START_REF] Regnerus | Linked lives, faith, and behavior: intergenerational religious influence on adolescent delinquency[END_REF].

Research in pre-adolescents is complicated by the fact that they start to internalize parental standards, like religious beliefs. This process of internalizing religious beliefs is known to be accompanied by a constructive phase of doubt, the so called 'quest' phase [START_REF] Batson | The religious experience: a social-psychological perspective[END_REF], which might be accompanied by some anxiety and conflict [START_REF] Kojetin | Quest: constructive search of religious conflict?[END_REF]. In general, as is known for physical maturation, boys also show a slower moral maturation than girls. Religious parents may, thus, meet resistance from boys at an older age [START_REF] Regnerus | Linked lives, faith, and behavior: intergenerational religious influence on adolescent delinquency[END_REF]. When dealing with healthy religious development, the proportion of internalized religious beliefs increases along with the development of an authentic personality. This leads to the integration of religion in the person's life and vice versa [START_REF] Vries-Schot | Healthy religiosity and salutary faith. Clarification of concepts from the perspective of psychology, psychiatry and of theology[END_REF]. The likelihood of a positive transmission of the parents´ religion to their children increases when the parents are actively religious [START_REF] Pearce | The impact of family religious life on the quality of mother-child relations[END_REF][START_REF] Kieren | Following the Leaders: Parent's Influence on Adolescent Religious Activity[END_REF] and when parents agree on the subject of religion. In this way the children do not receive mixed messages about the role of religion in life [START_REF] Myers | An interactive model of religiosity inheritance: the importance of family context[END_REF].

The mechanisms by which parents' religion affects mental health have not yet been conclusively identified. Several possibilities have been suggested [START_REF] George | Explaining the relationships between religious involvement and health[END_REF][START_REF] Josephson | Religion and spirituality in child and adolescent psychiatry: a new frontier[END_REF]. Religion could influence how individuals evaluate occurring life-events and stress, provide meaning, and determine a sense of responsibility and efficacy. Further, religion may entail religious coping skills and hence have a buffering effect on life stress. Next, religion mostly includes prescriptions for a healthy lifestyle, and the embedment in a religious community provides for social support and various psychosocial resources.

This study was designed to extend our knowledge about the role of religion in preadolescents and examine main and interactive effects of the religiosity of mother and parental harmony on the subject of religion on mental health problems of pre-adolescents. Since prior work on the relationship between religion and mental health in adolescents and adults led to mixed findings, we did not hypothesize a direction of the effect of maternal religiosity, but hypothesized that religious disharmony would be related to increased levels of problem behaviors. Finally, we hypothesized that the effects of religiosity and religious harmony would differ for internalizing and externalizing behavior problems, and that hits difference would vary across gender.

Methods

Study design

The Tracking Adolescents' Individual Lives Survey (TRAILS) is a prospective cohort study of Dutch pre-adolescents (age 10-12 years old). The key objective of TRAILS is to chart and explain the development of mental health from preadolescence into adulthood, both at the level of psychopathology and the levels of underlying vulnerability and environmental risks. As the incidence of emotional and behavioral problems increases substantially in adolescence, preadolescents are monitored until the age of 24 using biennial assessments. This report is based on data collected during the first wave of measurements, which ran from March 2001 until July 2002. The survey was approved by the national ethical committee 'Centrale Commissie Mensgebonden Onderzoek'. Informed consent was obtained from all parents after the nature of the study had been fully explained. Children were excluded from the study if they were incapable to participate due to mental retardation or a serious physical illness or handicap; or if no Dutch-speaking parent or parent surrogate was available, and it was not feasible to administer part of the measurements in the parent's language. Of all children approached for enrollment in the study (i.e., selected by the municipalities and attending a school that was willing to participate, n=3145), 6.7% were excluded because incapability or language problems. A detailed description of the sampling procedure and method is provided in de Winter et al. [START_REF] Myers | An interactive model of religiosity inheritance: the importance of family context[END_REF]. In total 2230 children (mean age = 11.09, SD = 0.55, 50.8% girls) were enrolled in the study (i.e. both child and parent agreed to participate). When looking into the groups of responders and nonresponders it was discovered that they did not differ with respect to gender, parental education, proportion of single-parent families, teacher-rated behavioral problems, or school absence. However, children in the non-response group did need additional help for learning difficulties more frequently [START_REF] De Winter | Evaluation of no-response bias in mental health determinants and outcomes in large sample of pre-adolescents[END_REF].

Data collection

Well-trained interviewers visited one of the parents or guardians (mothers, 95.6%) at their homes to administer an interview. They were asked to fill out a questionnaire. Children were measured at school. There they filled out questionnaires, in groups, under the supervision of one or more TRAILS assistants. Further, they were assessed individually. Teachers were asked to fill out a brief questionnaire for all participating TRAILS-children in their class. Measures used in the present study are described in more detail below.

Child mental health

To assess the child's mental health three measures were used: the parent-rated Child Behavior Checklist (CBCL) [START_REF] Achenbach | Manual for the Child Behavior Checklist / 4-18 and 1991 profile[END_REF], the Youth Self-Report (YSR) [START_REF] Achenbach | Manual for the Youth Self-Report and 1991 profile[END_REF] and the Teacher checklist of Psychopathology (TCP). The TCP is developed by the TRAILS team and contains nine descriptions of behaviors, based on the Teacher's Report Form [START_REF] Achenbach | Manual for the Teachers Report Form and 1991 profile[END_REF]. Response options for each description on the checklist range from 0 = not applicable to 4 = very clearly or frequently applicable. The 'socioeconomic status' variable (SES) was also addressed in TRAILS and is applied in this study as covariate. This item was measured as the average of five items: income level, educational level of both the father and the mother, and occupational level of each parent.

The lowest 25% of scores were considered as 'low SES', the highest 25% as 'high SES', and the rest was labeled as 'middle SES'. The internal consistency of these variables is satisfactory (Cronbach's alpha 0.84).

Religion

In TRAILS three descriptive questions regarding religion were answered by the mother about herself and her husband.. The first question was if the mother considered herself as a religious person, the second if she was affiliated with a church or denomination, and the third question was about church attendance. On the base of this information, three variables were constructed: religiosity of the mother, religiosity of the father and parental harmony on the subject of religion. After dichotomizing the three questions to 'yes'/ 'no' answers, parents were assigned to the category 'no religiosity' (no 'yes' answers), 'passive religiosity' (first and second question answered with 'yes', third question answered with '(almost) never') and 'active religiosity' (first and second question answered with 'yes', third question answered with 'monthly' or more).

Only in case of three missing values, 'religiosity mother' or 'religiosity father' was considered as missing. The variable 'religious harmony' was constructed by comparing the 'religiosity mother' variable with the 'religiosity father' variable, whereas totally similar categorization was considered as parental harmony, leaving all other options as parental disharmony. Data analysis The data were analyzed by four multivariate analyses of covariance (MANCOVA's, using the Generalized Linear Model module of SPSS, version 12.0), two with internalizing problem behavior (one for 'religiosity mother', one for 'religiosity father') and two with externalizing problem behavior as dependent variable (one for 'religiosity mother', one for 'religiosity father'). 'Between-subject factors were 'religiosity mother' (3 levels), 'religiosity father' (3 levels), 'religious harmony' (2 levels) and 'gender' (2 levels). 'Within-subject factors were rater (3 levels: self-report, YSR; parent-report, CBCL and teacher-report, TCP) and scale, with 3 levels for internalizing problem behavior (subscales: 'anxious/depressed', 'withdrawn/depressed' and 'somatic complaints') and 2 levels for externalizing problem behavior (subscales: 'aggression' and 'delinquency'). Socioeconomic status (SES) and parental divorce were included as covariates. Results were considered to be significant when p < 0.05. Forty-four cases were excluded from the analyses as they responded without information on religiosity.

Results

Descriptives of the dependent and independent variables are given in Table 1 & 2. Scores on internalizing and externalizing problem behavior are reported as mean item values. Active maternal religiosity was present in 21.7% of the sample. In the vast majority of the families (84.2%) there appeared to be religious harmony.

Table 1 & 2 about here

Internalizing problem behavior

The MANCOVA analysis with religiosity mother, religious harmony and gender as betweensubject factors, and rater and scale as within-subject factors aggregated over all raters and all subscales showed a marginally significant interaction effect of religiosity mother by harmony (F(2,1476) = 3.000, p = 0.050).

Since there were no interaction effects of the between-subject and within-subject factors, post-hoc analyses were performed on internalizing symptoms aggregated over all raters and all subscales. Basically, the interaction effect of the between-subject factors indicate that the influence of maternal religiosity was dependent on the presence or absence of religious harmony. This complex effect is illustrated in figure 1 for internalizing symptoms aggregated over all raters and all scales.

Figure 1 As shown in figure 1, the level of internalizing problem behavior was not significantly influenced by maternal religiosity, as long as parents are mostly in harmony on the subject of religion. However, when parents disagreed on the subject of religion, passive maternal religiosity was associated with significantly higher levels of internalizing problem behavior when compared with 'no maternal religiosity' (F1,183) = 3.987, p = 0.047), while active maternal religiosity was not associated with significantly higher levels of internalizing problem behavior when compared with 'no maternal religiosity' (F1,101) = 2.454, p = 0.120). When comparing harmonious and disharmonious parents, no significance was shown in case the mother was not religious (F(1,745) = 3.406, p = 0.065), passive religious (F(1,214) = 0.082, p = 0.774) or active religious (F(1,145) = 2.883, p = 0.092).

In a next set of post-hoc analyses we attempted to further characterize the interaction effects of religiosity mother by religious harmony in terms of rater and subscale. The YSR and TCP did not show any significant results of internalizing problem behavior. There was a significant effect of religiosity mother on the CBCL (F(6,3462)= 2.218, p = 0.039), reflecting higher internalizing problem scores in case of passive or active maternal religiosity compared to no maternal religiosity. The CBCL showed also a religiosity mother by harmony interaction effect (F(6,3462) = 2.275, p = 0.034). The scale 'withdrawn/depressed' showed a religiosity mother by harmony interaction effect (F(2,1752) = 5.272, p = 0.005), while the scale 'anxious/depressed' showed a religiosity mother effect (F(2,1753) = 3.277, p = 0.038), reflecting higher scores on the scale 'anxious/depressed' in case of passive or active religiosity. There was no main effect of religious harmony (p's > .05).

The MANCOVA analysis with religiosity father, religious harmony and gender as between-subject factors, and rater and scale as within-subject factors aggregated over all raters and all subscales showed no significant interaction effects, nor significant main effects .

In summary, the influence of maternal religiosity was dependent on the presence or absence of religious harmony. Internalizing symptoms increased significantly when parents disagreed on religion in combination with passive religious mothers when compared to not religious mothers. This effect was only found on the CBCL, and more specifically on the subscales withdrawn/depressed and anxious/depressed. Paternal religiosity had no significant effects on the level of internalizing symptoms in pre-adolescents.

Externalizing problem behavior

The MANCOVA analysis with religiosity mother, religious harmony and gender as betweensubject factors, and rater and scale as within-subject factor did not show any significant interaction effect. There was a main effect of gender (F(1,1495) = 111.251, p < 0.001) with boys having overall more externalizing problem behaviors than girls. There were several interaction effects of between-subject and within-subject factors, indicating that the effects of the betweensubject factors were specific to certain raters and/or subscales of externalizing problems. We found a harmony by scale effect (F(1,1495) = 5.297, p = 0.021), a harmony by rater by scale effect (F(2,1494) = 3.186, p = 0.042), and a maternal religiosity by rater by scale (F4,2990) = 2.995, p = 0.018). Further post-hoc analyses were performed to localize these interaction effects of harmony. Analyses on the YSR showed a main gender effect (F(1,1809) = 60.622, p < 0.001).

However, when testing boys separately, a significant main harmony effect was established (F(1,885) = 6.899, p = 0.00)) showing less externalizing symptoms in case of religious harmony, whereas analyses of the girls' scores did not show significant effects. This harmony effect for boys was significant on both subscales aggression and delinquency (respectively F(1,890) = 6.049, p = 0.014 and F(1,890) = 5.611, p = 0.018). Analyses on the CBCL and TCP did not show any significant results.

The MANCOVA analysis with religiosity father, religious harmony and gender as between-subject factors, and rater and scale as within-subject factor did not show a significant interaction effect. There was a main effect of gender (F(1,1495) = 112.134, p < 0.001) with boys having overall more externalizing problem behaviors than girls. There were several interaction effects of between-subject and within-subject factors, indicating that the effects of the betweensubject factors were specific to certain raters and/or subscales of externalizing problems. We found a religiosity father by scale effect (F(2,1495) = 3.239, p = 0.039) and a religiosity father by rater by scale effect (F(4,2990) = 2.990, p = 0.018).Further post-hoc analyses were performed to localize the religiosity father by scale interaction effect. However, no significant results were reached at the YSR, CBCL and TCP. In summary, boys reported less externalizing symptoms on the YSR in case of religious harmonious parents. Paternal religiosity had no significant effects on the level of internalizing symptoms in pre-adolescents.

Discussion

In this paper, the relationship between the religiosity of parents, religious harmony, and mental health problems among pre-adolescents has been examined. First, we hypothesized that religious disharmony would be related to increased levels of problem behavior. These data showed that the influence of maternal religiosity on internalizing symptoms was dependent on the presence or absence of religious harmony, and this was particularly apparent on the CBCL. Internalizing symptoms increased significantly when parents disagreed on religion in combination with passive religious mothers, compared to nonreligious mothers. Boys reported more externalizing symptoms on the YSR in case of religious disharmonious parents.

Religiosity of father did not influence the level of internalizing and externalizing symptoms. These findings are in line with our hypothesis.

Secondly, we hypothesized that effects of religiosity and religious harmony might differ for internalizing and externalizing behavior problems, and might differ for boys and girls. Effects of religiosity and religious harmony did differ for internalizing and externalizing behavior problems, in terms of main effects and interactions effects, as well as in gender and rater. Our data showed also that effects of religiosity and religious harmony were gender specific in case of externalizing behavior problems, but not in internalizing behavior problems. Our second hypothesis was confirmed because of these results.

We will now discuss these results more in detail in the following sections: differences in effects of maternal and paternal religiosity as well as religious harmony, gender effects and rater effects (respectively on the YSR, CBCL and TCP).

Considering the differences in effect of maternal and paternal religiosity as well as religious harmony, both internalizing and externalizing symptoms were influenced by religious harmony. Only in case of internalizing symptoms, passive religiosity of mother played a role, while religiosity of father did not play a role at all. Literature indicates that it is of importance which parent is active or passive religious in adolescents religiosity [START_REF] Kieren | Following the Leaders: Parent's Influence on Adolescent Religious Activity[END_REF]. Concerning the role of passive religious mothers in the mental health of pre-adolescents in this study, one could say that passive religiosity is associated with a less clear religious orientation and less religious commitment. By consequence children could be left with confusion on the matter of religion, which could result in more internalizing problem behavior. It could be that the pre-adolescents of the passive religious mothers have difficulty in finding a role model to identify with on the subject of religion. This corresponds with the developmental stage of preadolescents. At the age of 10-12 year, pre-adolescents are expected to be mostly dependent on the religious context of the parents, while they start with developing their own opinions and choosing their own ways. On the subject of religiosity, this development could play a role, for example in thinking about existential questions and in asking questions about religion. This beginning of the 'quest' phase might be accompanied by some anxiety and conflict [START_REF] Kojetin | Quest: constructive search of religious conflict?[END_REF]. In this sample, pre-adolescents of passive religious mothers and of parents who disagreed on the subject of religion, could have some extra difficulties because of the combination of a less clear religious orientation, possibly accompanied with confusing messages about the role of religion in life. Seen in this way, the pre-adolescents of this sample, in their specific age-related development, are more vulnerable to internalizing problem behavior because of some instability in their religious context. When parents are disharmonious on the subject of religion, this couldon its own -be a source of conflict and uncertainty, generating mental health problems [START_REF] Varon | Relationship between maternal church attendance and adolescent mental health and social functioning[END_REF][START_REF] Wong | A systematic review of recent research on adolescent religiosity/spirituality and mental health[END_REF][START_REF] Brody | Parental religiosity, family processes, and youth competence in rural, two-parent African American families[END_REF].

Gender effects were only reported on externalizing problem behavior, and only in relation to religious harmony of the parents. Our data showed that boys scored themselves as having more externalizing symptoms in case of religious disharmony. Regnerus [START_REF] Regnerus | Linked lives, faith, and behavior: intergenerational religious influence on adolescent delinquency[END_REF] showed that religious disharmony between parents eliminates most of the protective effect of religiosity of parents on delinquent behavior in adolescents.

Finally, different rater effects are being reported in this study. Boys reported themselves as having more externalizing symptoms in case of religious disharmony of their parents (YSR), while mothers reported more internalizing symptoms in case of the combination of passive religiosity of the mother and religious disharmony (CBCL). Abbotts et al. [START_REF] Abbotts | Is going to church good or bad for you? Denomination, attendance and mental health of children in West Scotland[END_REF] also reported different rater effects (children, parents and teachers). They found quite consistent results concerning children, parents and teachers in scoring aggression, but a lack of parental awareness of depression in their children. This was not the case in our study, as the mothers scored more internalizing symptoms (especially depression). A possible explanation for this difference could be that the mothers of the study of Abbotts et al. [START_REF]Diagnostic and statistical manual of mental disorders[END_REF] were weekly church attenders, while the mothers of our study were passive religious. This difference could reflect uncertainty in passive religious mothers, causing higher scores on internalizing behavior in their children.

We did not focus on the more general mechanisms in the domain of parenting and family functioning that may be involved in the associations between religiosity of parents and religious harmony and problem behaviors of their children. Religious attitudes and harmony may be correlated with proximal factors as parental personality characteristics, problem solving strategies, affective family climate and family interactions that influence childrens' adaptation and level of functioning.

These should be addressed in future studies.

This study has certain limitations. We could only use a global construct for religion. The 'religious harmony' variable is mainly based on the information provided by the mother. This could be the cause of some biases. All results displayed rather small effects, indicating that these effects only partly explain the observed variance. Although three covariates were included, more covariates and mediators concerning the religious context of pre-adolescents have as yet to be investigated.

This study is unique compared to other studies because of the age of the participants as well as the fact that this study is based on a community based sample. Religiosity of mother and harmony of parents on the subject of religiosity both turned out to influence internalizing problem behavior.

Externalizing problem behavior was related to religious disharmony between parents. From these results it becomes clear that religiosity is an important correlate in the mental health of pre-adolescents.

Further research needs to be conducted to investigate the religious context of pre-adolescents and to chart confounders and mediators in the relationship between religion and mental health. Longitudinal research is necessary to investigate the developmentally direction of the relationship between religion and mental health.

  Figure 1

Table 1 :

 1 Descriptives of the independent variables

	Independent variables

We are grateful to all pre-adolescents, their parents and teachers who participated in this research and to everyone who worked on this project and made it possible. This research is part of the Tracking Adolescents' Individual Lives Survey (TRAILS).

Description of financial support

TRAILS

has been financially supported by various grants from the Dutch Organization for Scientific Research NWO (Medical Research Council program grant GB-MW 940-38-011; ZonMW Brainpower grant 100-001-004; ZonMw Risk Behavior and Dependence grant 60-60600-97-118; ZonMw Culture and Health grant 261-98-710; Social Sciences Council mediumsized investment grant GB-MaGW 480-01-006 and project grants GB-MaGW 457-03-018 and GB-MaGW 452-04-314; NWO large-sized investment grant 175.010.2003.005); the Sophia Foundation for Medical Research (projects 301 and 393), the Dutch Ministry of Justice (WODC), and the participating universities.


Victor Kadankov
email: kadankov@voliacable.com

Tetyana Kadankova
email: tetyana.kadankova@uhasselt.be

Busy period, virtual waiting time and number of customers in G δ |M κ |1|B system

Keywords: Busy period, Time of the first loss of the customer, First exit time, Value of the overshoot, Linear component, Resolvent sequence Mathematics Subject Classification (2000) 60G40, 60K20

In this article, we determine the integral transforms of several twoboundary functionals for a difference of a compound Poisson process and a compound renewal process. Another part of the article is devoted to studying the abovementioned process reflected at its infimum. We use the results obtained to study a G δ |M κ |1|B system with batch arrivals and finite buffer in the case when δ ∼ ge(λ). We derive the distributions of the main characteristics of the queuing system, such as the busy period, the time of the first loss of a customer, the number of customers in the system, the virtual waiting time in transient and stationary regimes. The advantage is that these results are given in a closed form, namely, in terms of the resolvent sequences of the process.

Introduction

The queueing models with batch arrivals and a finite buffer of the G δ |M κ |1|B type arise in telecommunication networks, transportation systems and manufacturing systems. One of the most important performance issues of the queue with finite buffer is losses which are caused due to the buffer overflow, see [START_REF] Abramov | On losses in M X |GI |1|n queues[END_REF][START_REF] Lee | Busy periods of Poisson arrival queues with loss[END_REF]. In this article, we address the problem of losses by determining the Laplace transforms of the time of the first loss and the number of lost customers. In addition, we consider other performance characteristics such as the busy period and the virtual waiting time. We propose a probabilistic approach based on the solution of the boundary problems for a two-component Markov process which governs functioning of the above-mentioned queueing system.

Experience shows that finding the aforementioned characteristics is not an easy task. So far, the embedded Markov chain approach has been widely used for their analysis (see [START_REF] De Boer | The remaining service time upon reaching a high level in M|G|1 queues[END_REF] and references therein). The idea to use embedded Markov chains for studying queueing systems was introduced by Erlang in the 1930s. The main disadvantage of this approach is that the analysis of the evolution of the system is performed only on the states of the system which constitute a countable set. So far, major efforts of the researches were focused on establishing the equivalence between the stationary distribution of certain characteristics of the system and stationary distribution of the corresponding characteristics of the embedded Markov chain. It is clear that many characteristics cannot be determined based only on the states which constitute the countable set.

A two-component Markov process with continuous time appears naturally to mathematically describe the evolution of a queueing system. The first component of this process describes the phase state of the system, whereas the second component ensures the Markov property of the process.

The evolution of the number of customers in such systems is described by a process with two reflecting boundaries. In the general case, this process is a difference of two renewal processes. Reflections from the upper boundary are generated by the supremum (infimum) of the process. Reflections from the lower boundary govern the server's behavior. In general, such processes are not Markovian, but by adding a complementary linear component we obtain a Markov process which describes the functioning of the queueing system. Determining primary performance characteristics of the queueing system translates to studying boundary functionals of this governing process. We have followed this approach for the queueing systems G δ |M κ |1|B with a finite buffer in the case when δ ∼ ge(λ).

For the queueing systems of M κ |G δ |1|B and G δ |M κ |1|B type, the governing process is the difference of the compound Poisson process and the compound renewal process complemented with a linear component. The main two-boundary characteristic of this random process is the joint distribution of {χ, L, T }, i.e., of the first exit time from the interval, the value of the overshoot and the value of the linear component at this instant. Other two-boundary characteristics, such as the number of intersections of the interval, the joint distribution of the supremum, infimum and the position of the process, the sojourn time of staying inside the interval, the first entry time into the interval and also boundary characteristics of the reflected process can be derived in terms of this functional. In order to determine the Laplace transforms of the main two-boundary characteristic for the difference of a compound Poisson process and a compound renewal process [START_REF] Kadankov | A two-sided exit problem for a difference of a compound Poisson process and a renewal process with a discrete phase space[END_REF][START_REF] Kadankov | Exit problems for the difference of a Poisson process and a renewal process[END_REF], we employ the method suggested in [START_REF] Kadankov | On the distribution of the first exit time from an interval and the value of overshoot through the boundaries for processes with independent increments and random walks[END_REF] for general Lévy processes. The main contribution of this approach is that the joint distribution of {χ, L, T } is derived in terms of more simple joint distributions of the one-boundary functionals of the process which are known. For an overview of the existing results on the two-boundary problems, we refer to [START_REF] Ezhov | Two-boundary problems for a random walk[END_REF][START_REF] Picard | On the first meeting or crossing of two independent trajectories for some counting processes[END_REF]. For the stateof-the-art in queueing systems with batch arrivals and finite buffer, see [START_REF] Ohsone | Queues with finite waiting room[END_REF][START_REF] Pacheco | Consecutive customer losses in regular and oscillating M X |G|1|n systems[END_REF][START_REF] Perry | Busy period analysis for M|G|1 and G|M|1 type queues with restricted accessibility[END_REF][START_REF] Takagi | Queueing Analysis, A Foundation of Performance Evaluation[END_REF] and the references therein.

The rest of the article is structured as follows. In Sect. 1 we introduce the process and necessary notations. In Sect. 2 we consider the lower and upper one-boundary characteristics of the process. The main result is the solution of the two-sided exit problem for the underlying process. Section 3 deals with the process reflected at its infimum. We then consider the first passage time of the upper boundary, the distribution of the increments of the process and its asymptotic behavior. Finally, in Sect. [START_REF] De Boer | The remaining service time upon reaching a high level in M|G|1 queues[END_REF] we apply the results obtained in the previous sections to study primary performance characteristics of the queueing system G δ |M κ |1|B, such as the busy period, the time of the first loss of a customer, the number of the lost customers, the number of customers in the system and the virtual waiting time in transient and stationary regimes. In addition, we derive the above-mentioned characteristics for the partial case of G δ |M κ |1|B, namely when P[δ = 1] = 1.

Preliminaries

Let κ, δ ∈ N = {1, 2, . . . } be positive independent integer random variables, and η ∈ (0, ∞) be a positive random variable independent of κ, δ with the distribution function F (x) = P[η ≤ x], x ≥ 0. We will assume that Eκ, Eδ, Eη < ∞. Introduce the sequences {η, η n }, {κ, κ n }, {δ, δ n }, n ∈ N, of i.i.d. (inside of each sequence) variables and define the monotone sequences

η 0 (x) = 0, η 1 (x) = η x , η n+1 (x) = η x + η 1 + • • • + η n , n∈ N, (1)
κ 0 = 0, κ n = κ 1 + • • • + κ n ; δ 0 = 0, δ n = δ 1 + • • • + δ n ; n ∈ N,
where η x ∈ (0, ∞) is a random variable with the following distribution function

F x (u) = P[η x ≤ u] = F (x + u) -F (x) 1 -F (x) -1 , u≥ 0. Denote by {π(t)} t≥0 ∈ Z + = {0, 1, . . . } a compound Poisson process E z π(t) = e tk(z) , k(z)= μ E z κ -1 , |z| ≤ 1,
where μ > 0 is the intensity of the jumps and κ is a jump size. For all t ≥ 0 define a renewal process generated by the random sequence {η n (x)} n∈Z + :

N x (t) = max n ∈ Z + : η n (x) ≤ t ∈ Z + , x ≥ 0.
Introduce a right-continuous step process

D x (t) = π(t) -δ N x (t) ∈ Z = {0, ±1, . . . }, t ≥ 0; D x (0) = 0. (2)
Note, that inter-arrival times of the positive jumps are exponentially distributed with parameter μ, the positive jumps themselves are of a random size κ, and there occur negative jumps of size δ n at time instants η n (x), n ∈ N. We will call the process {D x (t)} t≥0 a difference of the compound Poisson process and a compound renewal process. Observe, that this process is not a Markov process in general. For all t ≥ 0, introduce a right-continuous linear component

η + x (t) = t + x, 0 ≤ t < η x , t -η N x (t) (x), t ≥ η x ∈ R + = [0, ∞), x ≥ 0. (3)
The process {η + x (t)} t≥0 increases linearly on the intervals [η n (x), η n+1 (x)), n ∈ Z + , it is killed to zero at the points η n (x), n ∈ N, and the value of the process at the instant t 0 ≥ η x is equal to the time elapsed from the moment of the last negative jump of Process (2) until t 0 . We will call Process (3) a linear component. By adding this linear component to the process {D x (t)} t≥0 , we obtain a right-continuous Markov process

{X t } t≥0 = D x (t), η + x (t) t≥0 ∈ Z × R + , ( 4
) X 0 = {0, x}, x ≥ 0,
which governs the process {D x (t)} t≥0 . The process defined in ( 4) is a Markov process. Note, that it is homogeneous with respect to the first component [START_REF] Ezhov | Markov processes homogeneous with respect to the second component[END_REF]. This fact will be used constantly when setting up the equations.

For all x ∈ R + , |z| ≤ 1, denote

fx (s) = Ee -sη x , f (s) = f0 (s), fx (s, z) = fx s -k(z) = E e -sη x z π(η x ) .
Lemma 1 (Kadankov et al. [START_REF] Kadankov | Intersections of an interval by a difference of a Poisson process and a compound renewal process[END_REF]) Let D x (t), η + x (t), t ≥ 0 be the random processes defined by (2)-(3), and ν s ∼ exp(s) be an exponential random variable independent of these processes. Then for all x ∈ R + , s > 0, |z| = 1, p ≥ 0 the following equality holds

D s x (z, p) = Ez D x (ν s ) e -pη + x (ν s ) = se -px s + p -k(z) 1 -fx (s + p, z) + s s + p -k(z) fx (s, z)Ez -δ 1 -f (s + p, z) 1 -f (s, z)Ez -δ . ( 5
)
In particular, for all x ∈ R + , s > 0, the following formulae are valid

D s x (z) = Ez D x (ν s ) = s s -k(z) + s fx (s, z) s -k(z) Ez -δ -1 1 -f (s, z)Ez -δ , |z| = 1. (6)
Here, and in the sequel, we will assume that the random variable δ ∈ N is geometrically distributed with parameter λ ∈ [0, 1):

P[δ = n] = (1 -λ)λ n-1 , n∈ N, Ez δ = z 1 -λ 1 -λz , |z| ≤ 1.
This assumption means that the process {D x (t)} t≥0 has geometrically distributed negative jumps which occur at time instants {η n (x)} n∈N . Here, and in the sequel, we will use the following short notation δ ∼ ge(λ). In this case, it is possible to obtain closed-form solutions for the one-and the two-sided boundary problems. In order to determine the main two-boundary characteristic of the process, we will require the one-boundary functionals of the process. These are the Laplace transforms of the joint distributions of the upper and lower one-boundary functionals of the process {X t } t≥0 . In the sequel we will use the following result.

Lemma 2 Let f (s) = Ee -sη . Then for s > 0 the equation

z -λ = (1 -λ) f s -k(z) , |z| < 1, λ∈ [0, 1) (7)
has a unique solution z = c(s) inside the circle |z| < 1. This solution is positive and c(s

) ∈ (λ, 1). If E[κ], E[η] < ∞, ρ = μ(1 -λ)E[κ] E[η], then for ρ > 1, lim s→0 c(s) = c ∈ (λ, 1); and for ρ ≤ 1, lim s→0 c(s) = 1.
A detailed proof of an analogous proposition for semi-continuous random walks can be found in the monograph of Spitzer [START_REF] Spitzer | Principles of Random Walk[END_REF]. The reasoning in that proof can be also applied to [START_REF] Kadankov | On the distribution of the first exit time from an interval and the value of overshoot through the boundaries for processes with independent increments and random walks[END_REF] (see also Lemma 1,[START_REF] Kadankov | Exit problems for the difference of a Poisson process and a renewal process[END_REF]).

One-and two-sided exit problems

Let X 0 = {0, x}, x ∈ R + , k ∈ Z + . Define τ k (x) = inf t : D x (t) < -k , T k (x) = -D x τ k (x) -k, inf{∅} = ∞,
i.e., the first undershoot time of the negative level -k by the process {D x (t)} t≥0 . We will use the convention that on the event

{τ k (x) = ∞} T k (x) = ∞. Denote B k (x) = {τ k (x) < ∞}.
We define the lower one-boundary functional of the process as follows

f k (x, m, s) = E e -sτ k (x) ; T k (x) = m, B k (x) , m∈ N.
This functional is determined by means of the following lemma. Lemma 3 (Kadankov and Kadankova [8]) Let {D x (t)} t≥0 be the difference of a compound Poisson process and a compound renewal process, δ ∼ ge(λ). Then (i) The Laplace transform of the joint distribution of {τ k (x), T k (x)}, k ∈ Z + , x ≥ 0 satisfies the following equality for s > 0, m ∈ N

f k (x, m, s) = fx s -k c(s) c(s) k (1 -λ)λ m-1 , ( 8
)
where c(s) ∈ (λ, 1) is the unique solution of [START_REF] Kadankov | On the distribution of the first exit time from an interval and the value of overshoot through the boundaries for processes with independent increments and random walks[END_REF]

inside the circle |z| < 1, fx (s) = E e -sη x , f (s) = E e -sη = f0 (s). (ii) If ρ > 1, then P[τ k (x) < ∞] = fx (-k(c)) c k < 1, and τ k (x), for all k ∈ Z + , x ≥ 0, is a defective random variable; if ρ ≤ 1, then P[τ k (x) < ∞] = 1, and τ k (x) is a proper variable for all k ∈ Z + , x ≥ 0.
Note that the value of the overshoot T k (x) does not depend on τ k (x) for any k ∈ Z + , and it is geometrically distributed: T k (x) ∼ ge(λ).

We now introduce a sequence which will be used to obtain the results in the sequel. The idea to employ this sequence for semi-continuous random walks and semicontinuous Lévy processes is due to Takács [START_REF] Takács | Combinatorial Methods in the Theory of Stochastic Processes[END_REF]. Since the function

fx s -k(z) = E e -sη x z π(η x ) = i∈Z + z i ∞ 0 e -st P η x ∈ dt, π(t) = i , |z| ≤ 1
is analytic inside the unit circle for all s, x ≥ 0, the function

Q s z (x) = (1 -λ) fx (s -k(z)) (1 -λ) f (s -k(z)) + λ -z , s,x ≥ 0, |z| < c(s) (9)
is analytic on the open set |z| < c(s). In this region, it can be represented as a power series

Q s z (x) = k∈Z + z k Q s k (x), s, x ≥ 0, |z| < c(s).
The coefficients of this expansion can be calculated by means of the inversion formula

Q s k (x) = 1 2πi |z|=α 1 z k+1 (1 -λ) fx (s -k(z)) (1 -λ) f (s -k(z)) + λ -z dz, α ∈ 0, c(s) . ( 10
)
We will call the sequence {Q s k (x)} k∈Z + , x ≥ 0, defined by Formula (10), the resolvent sequence of the process {D x (t)} t≥0 .

We now explain a probabilistic meaning of this sequence. Introduce a random sequence as follows (see [START_REF] Kadankov | Intersections of an interval by a difference of a Poisson process and a compound renewal process[END_REF]):

Y 0 (x) = 0, Y 1 (x) = π(η x ) -δ, Y n+1 (x) = Y 1 (x) + n i=1 Y i , Y n = X n (0), where Y = π(η) -δ ∈ Z, {Y, Y n }, n ∈ N, is a sequence of i.i.d. random variables.
Define a right-continuous step process in the following way:

R x (t) t≥0 = Y N x (t) (x) t≥0 ∈ Z, R x (0) = 0, x ∈ R + .
The sample paths of the process are constant on the time intervals [η n (x), η n+1 (x)), n ∈ Z + and they jump at the instants η n (x), n ∈ N. These jumps have the same distribution as Y .

= π(η)δ, where n ∈ {2, 3, . . . }, and Y 1 (x) . = π(η x )δ for n = 1. Here, and in the sequel, we will call the process {R x (t)} t≥0 a semi-Markov random walk generated by the sequences {η n (x)}, {Y n (x)}, n ∈ Z + . Let R + t = sup u≤t R 0 (u) be the supremum of {R 0 (t)} t≥0 . The generating function of R + t was found in [START_REF] Kadankov | Intersections of an interval by a difference of a Poisson process and a compound renewal process[END_REF]:

Ez R + νs = 1 -λ 1 -c(s) (1 -f (s))(z -c(s)) z -λ -(1 -λ) f (s -k(z)) , |z| ≤ 1,
where ν s is an exponential variable with parameter s > 0, independent from the process {R x (t)} t≥0 . It follows from [START_REF] Kadankov | Exit problems for the difference of a Poisson process and a renewal process[END_REF] and from the latter formula that for |z| < c(s)

Q s z (x) = 1 -c(s) 1 -f (s) fx (s -k(z)) c(s) -z Ez R + νs , |z| < c(s).
Comparing the coefficients of z k , k ∈ Z + , on both sides yields

Q s k (x) = 1 -c(s) 1 -f (s) k i=0 c(s) i-k-1 i j =0 E e -sη x , π(η x ) = j P R + ν s = i -j .
Denote by π s (η x ) ∈ Z + , s > 0, a random variable given by its distribution:

P π s (η x ) = k = 1 fx (s) E e -sη x , π(η x ) = k , k ∈ Z + .
Then the previous equality implies that

Q s k (x) = fx (s) 1 -f (s) 1 -c(s) c(s) k+1 k i=0 c(s) i P π s (η x ) + R + ν s = i , k ∈ Z + ,
which explains the probabilistic meaning of the resolvent sequence. Asymptotically, one has that

Q s k (x) ∼ c(s) -k as k → ∞. Let X 0 = {0, x}, x ≥ 0, k ∈ Z + ,
and introduce the upper one-boundary functionals of the process {X t } t≥0 :

τ k (x) = inf t : D x (t) > k , T k (x) = D x τ k (x) -k, η k (x) = η + x τ k (x) ,
i.e., the instant of the first crossing of the level k by the process {D x (t)} t≥0 , the value of the overshoot across the upper level and the value of the linear component η + x (•) at the instant of the first crossing (time since the last renewal). Denote B k (x) = {τ k (x) < ∞},

f k (x, dl, m, s) = E e -sτ k (x) ; η k (x) ∈ dl, T k (x) = m, B k (x) , m∈ N.
We will now determine the Laplace transforms of the upper one-boundary functionals of the process {D x (t)} t≥0 . Let k ∈ Z + and τ k = inf{t : π(t) > k}, T k = π( τ k )k be the first crossing time through the upper level k by the compound Poisson process {π(t)} t≥0 and the value of the overshoot at this instant. Denote by

ρ k (t) = P[π(t) = k], ∞ k=0 z k ρ k (t) = Ez π(t) = e tk(z) , |z| ≤ 1, p m k (dt) = P τ k ∈ dt, T k = m = μ k i=0 ρ i (t)P[κ = k -i + m] dt, m ∈ N.
Lemma 4 (Kadankov and Kadankova [START_REF] Kadankov | A two-sided exit problem for a difference of a compound Poisson process and a renewal process with a discrete phase space[END_REF]) Let {Q s k (x)} k∈Z + be the resolvent sequence of the process {D x (t)} t≥0 given by [START_REF] Kadankov | Intersections of an interval by a difference of a Poisson process and a compound renewal process[END_REF], k ∈ Z + , s,x ≥ 0. The following equalities hold:

f k (x, dl, m, s) = e -s(l-x) 1 -F (l) 1 -F (x) I{l > x}p m k d(l -x) + Φ s λ (0, dl, m) Q s k (x) -e -sl 1 -F (l) k i=0 Q s i (x)p m k-i (dl), ( 11
)
where

Φ s λ (0, dl, m) = e -sl [1 -F (l)] ∞ k=0 c(s) k p m k (dl); f k (x, s) = Ee -sτ k (x) = 1 -A k x (s) - s s -k(c(s)) Q s k (x) 1 -λ , ( 12
)
and

A k x (s) = k i=0 ρi (s)[1 -Q s k-i (x)(1 -λ) -1 ], ρk (s) = s ∞ 0 e -st ρ k (t) dt. For E[κ], E[η] < ∞ and ρ < 1, τ k (x) is a defective random variable and P τ k (x) < ∞ = 1 -(1 -ρ)(1 -λ) -1 Q k (x) < 1,
where {Q k (x)} k∈Z + , x ≥ 0 is the resolvent sequence of the process {D x (t)} t≥0 given by [START_REF] Kadankov | Intersections of an interval by a difference of a Poisson process and a compound renewal process[END_REF] for s = 0:

Q k (x) = 1 2πi |z|=α dz z k+1 (1 -λ) fx (-k(z)) (1 -λ) f (-k(z)) + λ -z , α ∈ 0, c(0) ; ( 13
) if ρ ≥ 1 then, for all k ∈ Z + and x ≥ 0, τ k (x) is a proper random variable.
Along with Expression (13) there exists another way to calculate Q k (x), which is more applicable from a practical point of view. We will now derive the recurrence formula for Q k (x). It follows from [START_REF] Kadankov | Exit problems for the difference of a Poisson process and a renewal process[END_REF] for s, z = 0 that

Q 0 (x) = (1 -λ) λ + (1 -λ)f 0 -1 f 0 (x),
where, for all

k ∈ Z + , f k (x) = P[π(η x ) = k] = ∞ 0 P[η x ∈ dt, π(t) = k], f k = f k (0). Again, it follows from (9) for s = 0 that (1 -λ) fx -k(z) = (1 -λ) f -k(z) Q z (x) + (λ -z)Q z (x).
Comparing the coefficients of z k , k ∈ N, on both sides implies that

(1 -λ)f k (x) = (1 -λ) k i=0 Q i (x)f k-i + λQ k (x) -Q k-1 (x).
Combining similar terms yields

λ + (1 -λ)f 0 Q k (x) = (1 -λ)f k (x) + Q k-1 (x) -(1 -λ) k-1 i=0 Q i (x)f k-i .
The latter formula is a recurrence relation which allows to calculate successively the terms Q k (x) given the previous terms Q 0 (x), . . . , Q k-1 (x). For instance, given the expression for Q 0 (x) one finds that

Q 1 (x) = 1 -λ λ + (1 -λ)f 0 f 1 (x) + 1 -(1 -λ)f 0 λ + (1 -λ)f 0 f 0 (x) . Denote by D - x (t) = inf [0,t] D x (•) the running infimum of the process on [0, t]. Theorem 1 Let r ∈ Z + , E - r (x, z, s) = E[z D x (ν s ) ; D - x (ν s ) ≥ -r], |z| ≤ 1 be the gen- erating function of the joint distribution of {D x (ν s ), D - x (ν s )}. Then (i) The generating function E - r (x, z, s) is such that E - r (x, z, s) = (1 -z)A z x (s) + (1 -z)f r (x, s) 1 -λ λ -z A z 0 (s)z -r , ( 14
)
where f r (x, s) = E[e -sτ r (x) ; B r (x)] = c x (s)c(s) r , c x (s) = fx (s -k(c(s)), A z x (s) = k∈Z+ z k A k x (s) = s s -k(z) 1 1 -z - 1 1 -λ Q s z (x) . (ii) The joint distribution E - r (x, u, s) = P[D x (ν s ) ≥ u, D - x (ν s ) ≥ -r], u ∈ [-r, ∞) satisfies the following equality E - r (x, u, s) = 1 -A u-1 x (s) -f r (x, s) 1 -EA u+r+δ-1 0 (s) , ( 15
)
where

EA k+δ x (s) = (1 -λ) i∈N λ i-1 A k+i x (s), A u x (s) = 0 for u < 0. (iii) Under Condition (A), namely, ρ = (1 -λ)μEηEκ = 1, (A) σ 2 = μ Eκ(κ -1) + EκEη 2 (1 -λ)(Eη) 2 < ∞,
the following limiting equality holds as B → ∞, r > 0,

P D x tB 2 ≥ [uB], D - x tB 2 ≥ [-rB] → 1 σ √ 2πt u+2r u e -v 2 /2σ 2 t dv,
where [a] is the integer part of the number a, u ∈ [-r, ∞).

Proof In accordance with the total probability law and Markov property of τ r (x) for all r ∈ Z + , we derive the following equation

Ez D x (ν s ) = E z D x (ν s ) ; D - x (ν s ) ≥ -r + f r (x, s) 1 -λ z -λ z -r Ez D 0 (ν s ) , |z| = 1.
To write this equation, we used the path decomposition principle. It means that the increments of the process D x (ν s ) on the interval [0, ν s ] are realized either on the sample paths which do not cross the negative level -r, or on the sample paths which do intersect the level -r, and further evolution of the process is its probabilistic replica on [0, ν s ]. From the latter equation, we find that

E - r (x, z, s) = D s x (z) -f r (x, s) 1 -λ z -λ z -r D s 0 (z).
Observe that the function which enters the left-hand side of the equation is analytic in {z : |z| ≤ 1}. Therefore, the right-hand side is also an analytic function for all |z| ≤ 1.

Employing [START_REF] Ezhov | Markov processes homogeneous with respect to the second component[END_REF] and the definition of the resolvent (9), we get

E - r (x, z, s) = (1 -z)A z x (s) + (1 -z)f r (x, s) 1 -λ λ -z A z 0 (s)z -r , |z| ≤ 1.
It is not difficult to establish the following equality

∞ u=-r z u E - r (x, u, s) = 1 -f r (x, s) 1 -z z -r - z 1 -z E - r (x, z, s), |z| ≤ 1.
In view of the latter and the previous equality, we find that

∞ u=-r z u E - r (x, u, s) = 1 -f r (x, s) 1 -z z -r -zA z x (s) -f r (x, s) 1 -λ λ -z A z 0 (s)z -r+1 .
Comparing the coefficients of z u , u ∈ [-r, ∞), and taking into account that A λ x (s) = 0, we obtain [START_REF] Nguyen-Ngoc | Some martingales associated to reflected Lévy processes[END_REF].

Denote ẽt r (x, u, B) = P[D x (tB 2 ) ≥ [uB], D - x (tB 2 ) ≥ [-rB]], r > 0, u ≥ -r. It is clear that lim B→∞ ∞ 0 e -st ẽt k (x, u, B) dt = 1 s lim B→∞ E s/B 2 [kB] x, [uB] .
To be able to perform asymptotic analysis and verify the third statement of the theorem we will use the following equalities (see [START_REF] Kadankov | Intersections of an interval by a difference of a Poisson process and a compound renewal process[END_REF] where they were derived)

c s/B 2 = 1 -B -1 √ 2s/σ + o B -1 , lim B→∞ B -1 Q s/B 2 [kB] (x) = 2 sinh(k √ 2s/σ ) σ √ 2sEη = lim B→∞ B -1 EQ s/B 2 δ+[kB] , ( 16
) lim B→∞ A [kB] x s/B 2 = 1 -cosh(k √ 2s/σ ) = lim B→∞ EA δ+[kB] 0 s/B 2 .
Employing these equalities and also [START_REF] Nguyen-Ngoc | Some martingales associated to reflected Lévy processes[END_REF], we find that lim Using this formula to invert the Laplace transforms in the previous equality, we obtain the limiting equality of the theorem.

B→∞ ∞ 0 e -st ẽt k (x, u, B) dt = s -1 I {u>0} e -u √ 2s/σ /2 -e -(2r+u) √ 2s/σ /2 + s -1 I {u∈[-r,0]} 1 -e u √ 2s/σ /2 -e -(2r+u) √ 2s/σ /2 , u≥ -r.
We will now consider the two-sided exit problem for the underlying process [START_REF] Avram | Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options[END_REF].

Let B ∈ Z + be fixed, k ∈ [0, B], r = B -k, X 0 = {0, x}, x ≥ 0, and introduce the random variable χ B r (x) = inf t : D x (t) / ∈ [-r, k] def = χ
as the first exit time from the interval [-r, k] by the process {D x (t)} t≥0 . This random variable takes values from a countable set {ξ n , n ∈ N} ∪ { η n (x), n ∈ N}, and it is a Markov time of the process {X t } t≥0 . (Here ξ n , n ∈ N, is a sequence of the jump times of the compound Poisson process.) Observe that exit from the interval can occur either through the upper boundary k, or through the lower boundary -r. In view of this remark, we introduce the events: A k = {D x (χ) > k}, i.e., the process {D x (t)} t≥0 exits the interval [-r, k] through the upper boundary k; A r = {D x (χ) < -r}, i.e., the process {D x (t)} t≥0 exits the interval [-r, k] through the lower boundary -r.

Denote by

T = D x (χ) -k I A k + -D x (χ) -r I A r , L = η + x (χ)I A k + 0 • I A r , P A k + A r = 1
the value of the overshoot through the boundaries of the interval [-r, k] by the process {D x (t)} t≥0 and the value of the linear component at the instant of the first exit. Here I A = I A (ω) is the indicator function of the event A. Denote

V k (x, dl, m, s) = E e -sχ ; L ∈ dl, T = m, A k , V r (x, m, s) = E e -sχ ; T = m, A r .
Theorem 2 (Kadankov and Kadankova [START_REF] Kadankov | A two-sided exit problem for a difference of a compound Poisson process and a renewal process with a discrete phase space[END_REF]) Let {D x (t)} t≥0 be the difference of the compound Poisson process and the renewal process

(2), δ ∼ ge(λ), Q s k def = Q s k (0). Then (i)
The Laplace transforms of the joint distribution of {χ, L, T } satisfy the following equalities for all x, s ≥ 0, m ∈ N,

V r (x, m, s) = Q s k (x) EQ s δ+B (1 -λ)λ m-1 , (17) V k (x, dl, m, s) = f k (x, dl, m, s) - Q s k (x) EQ s δ+B Ef δ+B (0, dl, m, s),
where the function f k (x, dl, m, s) is given by [START_REF] Kadankov | Some characteristics of the one line queuing system with losses[END_REF],

EQ s δ+B = k∈N (1 -λ)λ k-1 Q s k+B , Ef δ+B (0, dl, m, s) = k∈N (1 -λ)λ k-1 f k+B (0, dl, m, s).
(ii) For the Laplace transforms of the first exit time χ , the following formulae hold

E e -sχ ; A r = Q s k (x) EQ s δ+B , (18) E e -sχ ; A k = 1 -A k x (s) - Q s k (x) EQ s δ+B 1 -EA δ+B 0 (s) , where EA δ+B 0 (s) = k∈N (1 -λ)λ k-1 A k+B 0 (s).
It is worth noting that the Laplace transforms V r (x, m, s), V k (x, dl, m, s) were determined in [START_REF] Kadankov | A two-sided exit problem for a difference of a compound Poisson process and a renewal process with a discrete phase space[END_REF] for the case when δ ∈ N is a positive arbitrarily distributed random variable. The results of Theorem 1 were obtained as a corollary for δ ∼ ge(λ). In the following section we will employ this joint distribution to study the boundary characteristics of the process reflected from the boundaries.

Reflections from the boundary

Denote by D r

x (t) = r + D x (t), t ≥ 0, the process starting from r ∈ Z when η + x (0) = x ≥ 0. Let r ∈ Z + , and for all t ≥ 0 we define a right-continuous process reflected at the boundary 0 as follows:

D r 0 (x, t) = D r x (t) -min 0, inf [0,t] D r x (•) ∈ Z + , ( 19
)
D 0 r (x, 0) = r.
The reflection from the lower boundary 0 is generated by the infimum of the process D r x (t). Observe that the first hitting of 0 by the process D r 0 (x, t) occurs at time τ r (x). Subsequent time periods between the hitting times have the same distribution as τ 0 (0).

Note that reflections from the boundaries generated by the infimum (supremum) were introduced by Lévy for a standard Wiener process. Applying the symmetry principle and the mirror reflection principle, Lévy determined the distributions of the boundary functionals of the reflected standard Wiener process. It appears that these distributions are the limit distributions for the reflected process after an appropriate scaling of time and space.

Passage of the upper boundary

We will now determine the one-boundary functionals for Process [START_REF] Picard | On the first meeting or crossing of two independent trajectories for some counting processes[END_REF]. For B ∈ Z + , r ∈ [0, B], denote by

τ B r (x) = inf t : D r 0 (x, t) > B def = τ , T k r (x) = D 0 r (τ ) -B, L k r (x) = η + x (τ )
the first crossing time of the upper level B by the process D r 0 (x, t), the value of the overshoot and the value of the linear component at this instant.

Note that these boundary functionals were studied in [START_REF] Lotov | On limit theorems for the first exit time from a strip for stochastic processes[END_REF] for the reflected Lévy processes generated by the infimum (supremum). The reflected spectrally one-sided Lévy processes generated by the infimum (supremum) of the process were considered in [START_REF] Avram | Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options[END_REF][START_REF] Nguyen-Ngoc | Some martingales associated to reflected Lévy processes[END_REF]. An interesting application in queueing theory for the spectrally one-sided Lévy process reflected by its infimum was given in [START_REF] Bekker | Queues with delays in two-state strategies and Lévy input[END_REF].

Lemma 5 Let D r 0 (x, t) {t≥0} be the process reflected by infimum [START_REF] Picard | On the first meeting or crossing of two independent trajectories for some counting processes[END_REF].

Then (i) The Laplace transform v r x (dl, m, s) = E[e -sτ B r (x) ; L ∈ dl, T = m] of the joint distribution of {τ B r (x), L, T } satisfies the following equality for s > 0 v r x (dl, m, s) = V k (x, dl, m, s) + V r (x, s) 1 -V 0 (0, s) V B (0, dl, m, s), ( 20
)
where k = Br, V r (x, s) = ∞ m∈N V r (x, m, s), and the functions V k (x, dl, m, s), V r (x, m, s) are determined by [START_REF] Pacheco | Consecutive customer losses in regular and oscillating M X |G|1|n systems[END_REF]; in particular,

v r x (s) = Ee -sτ B r (x) = 1 -A k x (s) + Q s k (x) EA δ+B 0 (s) -A B 0 (s) EQ s δ+B -Q s B . ( 21
)
(ii) Under Condition (A), the following equality is valid

lim B→∞ P τ B rB (x)/B 2 > t = 4 π n∈Z + e -t 2 (π(n+ 1 2 )σ ) 2 2n + 1 sin k(2n + 1)π/2 , where r ∈ (0, 1), k = 1 -r. (iii) The random variable τ B r (x) is proper (P[τ B r (x) < ∞] = 1)
and

Eτ B r (x) = A k x + Q k (x) EA δ+B 0 -A B 0 EQ δ+B -Q B (0) < ∞,
where

Q k (x) = Q 0 k (x), EQ δ+B = EQ 0 δ+B , A k x = k i=0 ρ i 1 -(1 -λ) -1 Q k-i (x) , ρ i = ∞ 0 P π(t) = i dt < ∞.
Let us verify Formula [START_REF] Spitzer | Principles of Random Walk[END_REF]. It follows from the definition of the process D r 0 (x, t) (see [START_REF] Picard | On the first meeting or crossing of two independent trajectories for some counting processes[END_REF]), the total probability law and the Markov property of χ that the following equation is valid:

v r
x (dl, m, s) = V k (x, dl, m, s) + V r (x, s)v 0 0 (dl, m, s).

Letting x = r = 0 in this equation, we find that

v 0 0 (dl, m, s) = V B (0, dl, m, s) 1 -V 0 (0, s) -1 .
Substituting the expression for the function v 0 0 (dl, du, s) into the previous equation, we get Formula [START_REF] Spitzer | Principles of Random Walk[END_REF]. Formula (21) follows from [START_REF] Spitzer | Principles of Random Walk[END_REF] and [START_REF] Perry | Busy period analysis for M|G|1 and G|M|1 type queues with restricted accessibility[END_REF].

Lemma 6 Under Condition (A) and for k > 0 the next limiting equalities hold

lim B→∞ B -2 S s/B 2 [kB] (x) = 1 sEη cosh(k √ 2s/σ ) -1 = lim B→∞ B -2 ES s/B 2 δ+[kB] , lim B→∞ EQ s/B 2 [δ+kB] -Q s/B 2 [kB] = 2μEκ σ 2 cosh(k √ 2s/σ ), ( 22
) lim B→∞ B A [kB] 0 s/B 2 -EA [δ+kB] 0 s/B 2 = √ 2s (1 -λ)σ sinh(k √ 2s/σ ),
where

S s k (x) = k i=0 Q s i (x), S s k (x) = 0 for k < 0.
The proof of the lemma can be found in the Appendix. Formulae ( 21), ( 16) and the latter equalities imply, for

B → ∞, k ∈ (0, 1), r = 1 -k, that Ee -s τ B rB (x)/B 2 → cosh(k √ 2s/σ ) - 2 sinh(k √ 2s/σ ) σ √ 2sEη √ 2s (1 -λ)σ × sinh( √ 2s/σ ) σ 2 2μκ / cosh( √ 2s/σ ) = cosh(r √ 2s/σ ) cosh( √ 2s/σ ) .
It is obvious that

∞ 0 e -st lim B→∞ P τ B rB (x)/B 2 > t dt = 1 s - 1 s cosh(r √ 2s/σ ) cosh( √ 2s/σ ) .
Inverting the Laplace transforms on both sides of the latter equality, we get the limiting equality of the theorem.

Increments of the process reflected at its infimum

Let r ∈ Z + , and denote by

D 0 -r (x, t) = D x (t) -min{0, inf [0,t] D x (•) + r} ∈ [-r, ∞
), the process reflected from the lower boundary -r. The reflections are generated by its infimum. Introduce

P s r (x, z) = Ez D 0 -r (x,ν s ) , |z| ≤ 1, p s r (x, u) = P D 0 -r (x, ν s ) ≥ u , u∈ [-r, ∞),
i.e., the generating function and the distribution function of the increments of the process sampled at the exponential time ν s . The following statement holds.

Theorem 3 Let {D 0 -r (x, t)} t≥0 be the process reflected at the lower boundary. Then

(i)
The following equalities are valid for r ∈ Z + , x ≥ 0, u ∈ [-r, ∞),

P s r (x, z) = (1 -z) A z x (s) + z -r f r (x, s) 1 -λ 1 -c(s) 1 -z λ -z A z 0 (s) , ( 23
) p s r (x, u) = 1 -A u-1 x (s) - 1 -λ 1 -c(s) f r (x, s) A u+r-1 0 (s) -EA δ+u+r-1 0 (s) ,
where A u x (s) = 0, for u < 0. (ii) Under Condition (A), for r > 0, u ≥ -r,

lim B→∞ P D 0 [-rB] x, tB 2 ≥ [uB] = 1 - 1 σ √ 2πt u+2r -u e -v 2 /2σ 2 t dv. ( 24
) (iii) If ρ = (1 -λ)μEηEκ < 1, then the ergodic distribution p r (u) = lim t→∞ P D 0 -r (x, t) ≥ u
exists, and the limiting equality

p r (u) = 1 - 1 -ρ Eη A u+r-1 0 -EA δ+u+r-1 0 holds, where A k x = k i=0 ρ i [1 -(1 -λ) -1 Q k-i (x)], ρ i = ∞ 0 P[π(t) = i] dt.
Proof In view of the total probability law, Markov property of τ r (x) and homogeneity property of the process with respect to the first component, we can write the following equation for the function P s r (x, z):

P s r (x, z) = E - r (x, z, s) + sf r (x, s)z -r s + μ + μf r (x, s)z -r 1 -f (s + μ) ∞ 0 i∈N a i (dl)P s i (l, z)z i , ( 25
)
a i (dl) = e -l(s+μ) 1 -F (l) P[κ = i] dl,
where the generating function

E - r (x, z, s) = E[z D x (ν s ) ; D - x (ν s ) ≥ -r],
|z| ≤ 1, is determined by [START_REF] Lotov | On limit theorems for the first exit time from a strip for stochastic processes[END_REF].

This equation reflects the following fact. The increments of the process {D 0 -r (x, t)} can take place either on the sample paths which do not hit the lower boundary -r, (the first term on the right-hand side), or on the sample paths which hit the lower boundary -r and stay there (the second term), or finally, on the sample paths which hit the boundary -r and then are reflected from the boundary (the third term). Denote

X(s, z) = μ 1 -f (s + μ) ∞ 0 i∈N a i (dl)P s i (l, z)z i .
Setting x = 0 in (25) and performing necessary calculations, we get

X(s, z) = 1 -λ 1 -c(s) μ ∞ 0 r∈N a r (dx)E - r (x, z, s)z r + s 1 -f (s + μ) s + μ - s s + μ .
Inserting this expression for the function X(s, z) into (25) yields

P s r (x, z) = E - r (x, z, s) + f r (x, s)z -r 1 -λ 1 -c(s) μ ∞ 0 r∈N a r (dx)E - r (x, z, s)z r + s 1 -f (s + μ) s + μ .
Inserting Expression [START_REF] Lotov | On limit theorems for the first exit time from a strip for stochastic processes[END_REF] for the function E - r (x, z, s) into the latter equality and performing necessary calculations, we find that

P s r (x, z) = (1 -z) A z x (s) + z -r f r (x, s) 1 -λ 1 -c(s) 1 -z λ -z A z 0 (s) .
It is not difficult to derive the following relation

ps r (x, z) = ∞ u=-r z u p s r (x, u) = z -r 1 -z - z 1 -z P s r (x, z), |z| ≤ 1.
The latter and the previous equality imply that

ps r (x, z) = z -r 1 -z -zA z x (s) -z -r+1 f r (x, s) 1 -λ 1 -c(s) 1 -z λ -z A z 0 (s).
Comparing the coefficients of z u , u ∈ [-r, ∞), we get the second formula of (23). We now verify (24). The first formula of ( 23) and ( 8) imply that

lim B→∞ f [rB] x, s/B 2 = e -r √ 2s/σ , r >0. Denote pt r (x, u, B) = P[D 0 [-rB] (x, tB 2 ) ≥ [uB]], r > 0, u ≥ -r. It is obvious that lim B→∞ ∞ 0 e -st pt k (x, u, B) dt = 1 s lim B→∞ p s/B 2 [rB] x, [uB] .
Employing [START_REF] Ohsone | Queues with finite waiting room[END_REF] and Lemma 6, we obtain lim

B→∞ ∞ 0 e -st pt k (x, u, B) dt = s -1 I {u>0} e -u √ 2s/σ /2 + e -(2r+u) √ 2s/σ /2 + s -1 I {u∈[-r,0]} 1 -e u √ 2s/σ /2 + e -(2r+u) √ 2s/σ /2 , u≥ -r.
In view of the formula s -1 e -a √ 2s/σ = 2 ∞ 0 e -st P[w t ≥ a] dt, we invert the Laplace transforms on the right-hand side of this equality, which yields the limiting equality of the theorem. For ρ < 1 the mathematical expectation of τ r (x) is finite. It follows from [START_REF] Kadankov | A two-sided exit problem for a difference of a compound Poisson process and a renewal process with a discrete phase space[END_REF] that

Eτ r (x) = Eη x + r(1 -λ)Eη (1 -ρ) -1 < ∞.
Moreover, the process D 0 -r (x, t) is of regenerative type [START_REF] Kuznetsov | Random Processes[END_REF]. The instants of the passages of the lower boundary are the regeneration times. Due to this property (see [START_REF] Kuznetsov | Random Processes[END_REF]) there exists an ergodic distribution of the process p k (u) = lim t→∞ P[D k 0 (x, t) ≤ u]. To determine this distribution it suffices to apply the Tauberian theorem to Formula (23). Here p k (u) = lim s→0 p s k (x, u).

Let {D 0 -r (x, t)} t≥0 be the process reflected from the lower boundary -r, r, k ∈ Z + . Introduce the following random variables

τ r,k (x) = inf t : D 0 -r (x, t) > k = τ , T r,k (x) = D 0 -r (x, τ ) -k, L r,k (x) = η + x (τ ),
i.e., the first exit time from the interval [-r, k] by the process D 0 -r (x, t), the value of the overshoot through the upper boundary k and the value of the linear component at this instant. Since the process X t is homogeneous with respect to the first component, then the random variables {τ r,k (x), T r,k (x), L r,k (x)} have the same distributions as {τ B r (x), T B r (x), L r (x)}, B = k + r. And hence their joint distribution is given by [START_REF] Spitzer | Principles of Random Walk[END_REF].

Theorem 4 Let {D 0 -r (x, t)} t≥0 be the process reflected from the lower boundary -r.

Denote by p s r,k (x, u)

= P[D 0 -r (x, ν s ) ≤ u; τ r,k (x) > ν s ], u ∈ [-r, k],
the distribution of the increments of the process sampled at the exponential time ν s , s > 0, given that the event {τ r,k (x) > ν s } takes place. Then

(i) The following equality is valid for r, k ∈ Z + , u ∈ [-r, k], p s r,k (x, u) = A u x (s) -Q s k (x) EA δ+u+r 0 (s) -A u+r 0 (s) EQ s B+δ -Q s B . (26)
(ii) Under Condition (A), the limiting equality

lim B→∞ P D [kB] x tB 2 ≤ [uB]; τ [rB],[kB] (x) > tB 2 def = p(t) = 4 π n∈Z + e -t 2 (π(n+ 1 2 )σ ) 2 2n + 1 × sin (r + u) n + 1 2 π cos r n + 1 2 π (27) holds, where r ∈ (0, 1), k = 1 -r, u ∈ [-r, k].
Proof Introduce the generating function

P s r,k (x, z) = E z D 0 -r (x,ν s ) ; τ r,k (x) > ν s .
According to the total probability law, the homogeneity property of the process X t with respect to the first component, Markov property of τ r,k (x), and properties of the exponential variable ν s , we can write the following equation

P s r (x, z) = P s r,k (x, z) + ∞ 0 m∈N v r x (dl, m, s)P s m+B (l, z)z m+k , |z| ≤ 1,
where v r x (dl, m, s) = E[e -sτ B r (x) ; L ∈ dl, T = m] is determined by ( 20), and

P s r (x, z) = E[z D 0 -r (x,ν s )
] is found in Theorem 3 by (23). This equation is written using the path decomposition principle. In other words, the increments of the process D 0 -r (x, ν s ) can be realized on one of the following self-excluding events: (1) the sample paths do not cross the upper boundary k, (2) the sample paths do intersect the upper boundary and then the evolution of the process is just a probabilistic replica of the process on [0, ν s ]. Inserting Expression (23) for the function P s r (x, z) into the latter equation, we find that

1 1 -z P s r,k (x, z) = A z x (s) -z k ∞ 0 ṽr x (dl, z, s)A z l (s) + z -r A z 0 (s) 1 -λ 1 -c(s) 1 -z λ -z × f r (x, s) - m∈N ∞ 0 v r x (dl, m, s)f m+B (l, s) , ( 28
)
where ṽr x (dl, z, s) = E[e -sτ B r (x) z T ; L ∈ dl]. Performing necessary calculations and taking into account ( 8) and [START_REF] Spitzer | Principles of Random Walk[END_REF]

yields m∈N ∞ 0 v r x (dl, m, s)f m+B (l, s) = f r (x, s) - 1 -c(s) 1 -λ Q s k (x) EQ s B+δ -Q s B .
In view of this equality and (28), we find that

P s r,k (x, z) -z k+1 (1 -v r x (s)) 1 -z = A z x (s) -z k ∞ 0 ṽr x (dl, z, s)A z l (s) - z k+1 (1 -v r x (s)) 1 -z + z -r A z 0 (s) 1 -z λ -z Q s k (x) EQ s B+δ -Q s B , |z| ≤ 1, ( 29
)
where ṽr

x (s) = E[e -sτ B r (x) ]. Then it is easily verified that 1 1 -z P s r,k (x, z) - z k+1 1 -z 1 -v r x (s) = k u=-r z u p s r,k (x, u).
Comparing the coefficients of z u , u ∈ [-r, k], on both sides of (29), we obtain (26).

For r ∈ (0, 1),

k = 1 -r, u ∈ [-r, k], denote p t r,k (x, u, B) = P D [kB] x tB 2 ≤ [uB]; τ [rB],[kB] (x) > tB 2 .
Employing Formula (26) and the limiting equality of Lemma 6, we find that

1 s lim B→∞ p s/B 2 [rB],[kB] x, [uB] = lim B→∞ ∞ 0 e -st p t r,k (x, u, B) dt = 1 -cosh(u + √ 2s/σ ) s + 1 s sinh(k √ 2s/σ ) cosh( √ 2s/σ ) sinh (u + r) √ 2s/σ def = p * (s), ( 30
)
where u + = max{0, u}. When u ∈ [-r, 0], we derive from this formula that

p * (s) = 1 s sinh(k √ 2s/σ ) cosh( √ 2s/σ ) sinh (u + r) √ 2s/σ , u∈ [-r, 0].
It is clear that s = 0 is not a singular point (pole or point of branching) of the function p * (s). On the half-plane (s) < 0, this function has simple poles at

s n = - 1 2 σ 2 π 2 n + 1 2 2 , n∈ Z + ,
and it is analytic on the whole plane apart from these points. Hence, for α > 0,

p(t) = 1 2πi α+i∞ α-i∞ e st p * (s) ds = n∈Z +

Res s=s n p * (s).

Calculating the residues of the function p * (s) at s n , we obtain the right-hand side of Formula (27) for u ∈ [-r, 0]. On can see that for u ∈ (0, k] the first term on the righthand side of (30) is analytic on the entire plane. Applying the inversion formula, we find that the contour integral of this term is equal to zero. The second term of (30) is the same also for u ∈ [-r, 0]. Thus, Formula (27) holds for u ∈ [-r, k].

Applications for the G δ |M κ |1|B system

Before considering the queueing system of interest, we stress the following facts.

Remark 1 Let B ∈ Z + be fixed, k ∈ [0, B], r = B -k. Introduce the process D B+1 k+1 (x, t) = -D 0 -r (x, t) + k + 1 ∈ [-∞, B + 1], (31) D B+1 k+1 (x, 0) = k + 1.
This process is reflected at the upper boundary B + 1, due to the infimum of the process D x (t). Introduce the following random variable:

τ k+1 (x) = inf t : D B+1 k+1 (x, t) < 1 = inf t : D 0 -r (x, t) > k = inf t : D r 0 (x, t) > B = τ B r (x).
This defining chain of stochastic equalities implies that τ k+1 (x) and τ B r (x) are identically distributed. And hence, by [START_REF] Takács | Combinatorial Methods in the Theory of Stochastic Processes[END_REF],

v k x (s) = Ee -sτ k (x) = 1 -A k-1 x (s) + Q s k-1 (x) EA δ+B 0 (s) -A B 0 (s) E Q s δ+B -Q s B , k ∈ [1, B + 1].
Remark 2 Denote by p s k+1,x (u

) = P[D B+1 k+1 (x, ν s ) ≥ u; τ k+1 (x) > ν s ], u ∈ [1, B + 1]
, the Laplace transform of the increments of the process D B+1 k+1 (x, t) on the event {τ k+1 (x) > t}. The definition of the process and Remark 1 imply that

P D B+1 k+1 (x, ν s ) ≥ u; τ k+1 (x) > ν s = P D 0 -r (x, ν s ) ≤ k + 1 -u; τ B r (x) > ν s .
It follows from the latter equality and from (26

) that for k, u ∈ [1, B + 1] p s k,x (u) = A k-u x (s) -Q s k-1 (x) EA δ+B+1-u 0 (s) -A B+1-u 0 (s) EQ s B+δ -Q s B . ( 32
)
We now introduce the process describing the functioning of the queueing system.

Let B ∈ Z + , k ∈ [0, B + 1], x ≥ 0. Define the Markov process Y k,x (t) = d k,x (t), η + x (t) ∈ [0, B + 1] × R + , Y k,x (0) = (k, x)
by means of the following stochastic recurrences

Y k,x (t) = (D B+1 k (x, t), η + x (t)), 0 ≤ t < τ k (x), Y 0,η + x (τ k (x)) (t -τ k (x)), t ≥ τ k (x), k ∈ [1, B + 1], Y 0,x (t) = ⎧ ⎪ ⎨ ⎪ ⎩ (0, η + x (t)), 0 ≤ t < η x , Y k,0 (t -η x ) with probability (1 -λ)λ k-1 , k = [1, B], t ≥ η x , Y B+1,0 (t -η x ) with probability λ B , t≥ η x .
The process Y k,x (t) {t≥0} serves as a mathematical model of the functioning of the G δ |M κ |1|B system with (δ ∼ ge(λ)). Let us describe how this system works.

(i) Customers arrive into the system in batches according to the renewal process N x (t) {t≥0} . The number of customers in every batch is a random variable identically distributed as δ ∼ ge(λ) ∈ N. (ii) The system has a finite buffer whose size is equal to B + 1 < ∞. Suppose that upon arrival of a new customer of size δ, there are k ∈ [0, B + 1] occupied places in the waiting room. Then min{r, δ} customers join the queue, and loss of size max{0, δ -r} occurs, where r = B + 1k is the size of the empty space in the buffer;

(iii) The duration of service completion is exponentially distributed with parameter μ > 0. Suppose that at time t the service cycle is accomplished. Then the occupied space in the buffer is reduced by min{k, κ}, where k ∈ [1, B + 1] is the number of the occupied places in the waiting room at time t -0. If at the instant of the service completion k -min{k, κ} > 0, then a new service cycle starts. If at the instant of the service completion k -min{k, κ} = 0, then the new service cycle starts upon arrival of a new customer (batch of customers).

For all t ≥ 0, the event {Y k,x (t) = (i, y)}, i ∈ [1, B + 1], y ≥ 0, means that at time t there are i customers in the buffer and that time y has elapsed since the last arrival up to time t. We assume that (k, x) is an initial state of the system. The event {Y k,x (t) = (0, y)} means that at time t the buffer is empty and the system is idle, and time y has elapsed since the last customer's arrival (up to time t). Hence, η y is the duration of the idle period (state (0, y)).

Thus, d k,x (t) is the number of customers in the buffer at time t, η + x (t) is the time elapsed since the last arrival of the batch up to time t. Note that the definition of the process Y k,x (t) (homogeneity of the process X t with respect to the first component) implies that the linear component η + x (t) does not depend on k. In the next part of this paper, we will determine the Laplace transforms of the main performance characteristics of the system. The Laplace transform of the busy period of the system is the subject of the next subsection.

5.1 Busy period of the system Suppose that the system starts functioning at time t 0 = 0 from the state (k, x), where k ∈ [1, B + 1] is the number of customers in the buffer, x ≥ 0 is the time elapsed since the last arrival up to time t 0 = 0. Denote by

b k (x) = inf t : d k,x (t) = 0 , η(x)= η + x b k (x)
the instant at which the system becomes empty for the first time and the value of the linear component at time b k (x), respectively. Hence, the interval [0, b k (x)] is a busy period of the (k, x)-type.

Corollary 1 Let b s k (x) = E[e -sb k (x) ; b k (x) < ∞]
be the Laplace transform of the busy period of the (k, x)-type. Then (i) The following equality holds

b s k (x) = 1 -A k-1 x (s) + Q s k-1 (x) EA δ+B 0 (s) -A B 0 (s) EQ s δ+B -Q s B , k ∈ [1, B + 1], ( 33
)
the random variable b k (x) is proper (P[b k (x) < ∞] = 1)
, and it has a finite mathematical expectation given by

Eb k (x) = A k-1 x -Q k-1 (x) EA δ+B 0 -A B 0 EQ δ+B -Q B < ∞. ( 34
) (ii) The Laplace transform b s k (x, dy) = E[e -sb k (x) ; η(x) ∈ dy] of the joint distribu- tion of {b k (x), η(x)} is such that for k ∈ [1, B + 1] b s k (x, dy) = f k-1 (x, dy, s) -Q s k-1 (x)
Ef δ+B (0, dy, s)f B (0, dy, s)

EQ s δ+B -Q s B , (35)
where the function

f k (x, dy, s) = E[e -sτ k (x) ; T k (x) ∈ dy, B k (x)], k ∈ Z + , is determined by (26).
Proof Formula (33) follows straightforwardly from ( 21) and Remark 1. Equalities ( 20) and ( 17) imply (35).

Time of the first loss of a customer

Suppose that the initial state of the system is

(k, x), k ∈ [0, B + 1], x ≥ 0. Introduce l k (x)
as time of the first loss of a customer (group of customers); i k,x (t) as the number of the lost customers on the time interval [0, t]; and i k,x = i k,x (l k (x)) as the number of the lost customers at time l k (x).

Corollary 2 Let l s k (x) = E[e -sl k (x) ; l k (x) < ∞]
be the Laplace transform of l k (x). Then the following relations hold

l s k (x) = fx (s) + (1 -f (s))S s k-1 (x) f (s) + (1 -f (s))E S s δ+B , k ∈ [0, B + 1], (36) l s k (x, m) = E e -sl k (x) ; i k,x = m = l s k (x)(1 -λ)λ m-1 , m∈ N, where S s k (x) = k i=0 Q s i (x), S s k (x) = 0 for k < 0.
The random variable l k (x) is proper with a finite mathematical expectation

El k (x) = Eη x -Eη + Eη ES δ+B -S k-1 (x) < ∞, where S k (x) = S 0 k (x), ES δ+B = ES 0 δ+B . Proof The functions l s k (x), k ∈ [1, B + 1]
, l s 0 (y) obey the following system of equations:

l s k (x) = V B+1-k (x, s) + ∞ 0 V k-1 (x, dy, s)l s 0 (y), ( 37
)
l s 0 (y) = fy (s)λ B+1 + fy (s) B+1 k=1 (1 -λ)λ k-1 l s k (0),
where the functions V r (x, s), V k (x, dy, s) are given by ( 17) and [START_REF] Perry | Busy period analysis for M|G|1 and G|M|1 type queues with restricted accessibility[END_REF]. Substituting the expression for the function l s 0 (y) from the second equation into the first, we get

l s k (x) = V B+1-k (x, s) + λ B+1 ∞ 0 V k-1 (x, dy, s) fy (s) + ∞ 0 V k-1 (x, dy, s) fy (s) B+1 k=1 (1 -λ)λ k-1 l s k (0).
Letting x = 0 in the latter equation, we find for the function

X(s) = B+1 k=1 (1 - λ)λ k-1 l s k (0) that X(s) + λ B+1 = λ B+1 + VB (λ, s) 1 - ∞ 0 V B (λ, dy, s) fy (s) , ( 38
)
where

VB (λ, s) = (1 -λ) B+1 k=1 λ k-1 V B+1-k (0, s), V B (λ, dy, s) = (1 -λ) B+1 k=1 λ k-1 V k-1 (0, dy, s).
Employing Formulae ( 11), ( 17), ( 18) and performing necessary calculations, we obtain

λ B+1 + VB (λ, s) = (1 -λ) EQ s δ+B -1 , 1 - ∞ 0 V B (λ, dy, s) fy (s) = (1 -λ)S B (λ, s) EQ s δ+B -1 ,
where

S B (λ, s) = f (s) + (1 -f (s))ES s δ+B .
These equalities, Formula (38) and the second equality of (37) imply that l s 0 (y) = fy (s)S B (λ, s) -1 .

Inserting the right-hand side of this equality into the second equality of (37), we get

l s k (x) = Q s k-1 (x) EQ s δ+B + S B (λ, s) -1 ∞ 0 V k-1 (x, dy, s) fy (s), k ∈ [1, B + 1].
In view of ( 11), ( 17) and ( 18), we find that

∞ 0 V k-1 (x, dy, s) fy (s) = fx (s) + 1 -f (s) S s k-1 (x) - Q s k-1 (x) EQ s δ+B S B (λ, s).
The latter and the previous equality imply the first equality of (36). We now verify the second equality. Observe that i k,x ∼ ge(λ). This can be formally derived from the first formula of (17) and from the following system of equations

l s k (x, m) = V B+1-k (x, s)(1 -λ)λ m-1 + ∞ 0 V k-1 (x, dy, s)l s 0 (y, m), l s 0 (y, m) = fy (s)(1 -λ)λ B+m + fy (s) B+1 k=1 (1 -λ)λ k-1 l s k (0, m).
To solve this system, one can apply a similar reasoning as for the system (37).

Corollary 3 Let ν s ∼ exp(s) be an independent from the process Y k,x (t) exponential variable with parameter s > 0. Denote by I s k,x (n) = P[i k,x (ν s ) = n], n ∈ Z + , the distribution of the number of the lost customers on the time interval [0, ν s ]. For all k ∈ [1, B + 1], x ≥ 0, the following equalities are valid

I s k,x (0) = 1 -l s k (x), ( 39
) I s k,x (n) = l s k (x)(1 -λ) 1 -l s B+1 (0) λ + (1 -λ)l s B+1 (0) n-1 , n∈ N. Proof Let Ĩ s k,x (z) = E[z i k,x (ν s ) ],
|z| ≤ 1, be the generating function of the distribution of the number of the lost customers. Then it obeys the equation

Ĩ s k,x (z) = 1 -l s k (x) + Ls k,x (z) Ĩ s B+1,0 (z), (40)
where (see (36))

Ls k,x (z) = E e -sl k (x) z i k,x = l s k (x)z 1 -λ 1 -zλ .
Letting k = B + 1, x = 0 in (40), we find that

Ĩ s B+1,0 (z) = 1 -l s B+1 (0) 1 -Ls B+1,0 (z) -1 .
Inserting the right-hand side of this equality into (40) implies

Ĩ s k,x (z) = 1 -l s k (x) 1 -z 1 -zλ -z(1 -λ)l s B+1 (0)
.

Comparing the coefficients of z n , n ∈ Z + , we obtain (40) of the corollary.

Number of customers in the system

Let ν s ∼ exp(s) be an exponential r.v. with parameter s > 0. Introduce the transient probabilities of the process

d k,x (t) {t≥0} , k ∈ [0, B + 1], x ≥ 0, q s k,x (0) = P d k,x (ν s ) = 0 , q s k,x (u) = P d k,x (ν s ) ≥ u , u∈ [1, B + 1].

Theorem 5

The distribution of the number of customers in the system sampled at the exponential time ν s is such that

q s k,x (0) = 1 -A k-1 x (s) -A B 0 (s) -EA δ+B 0 (s) 1 -λ EQ s δ+B C s k-1 (x), (41) q s k,x (u) = A k-u x (s) + A B+1-u 0 (s) -EA δ+B+1-u 0 (s) 1 -λ EQ s δ+B C s k-1 (x),
where

A k x (s) = S s k (x) = 0 for k < 0, C s k (x) = fx (s) 1 -f (s) -1 + S s k (x).
Under Condition (A), the following limiting equality holds

lim B→∞ P d [kB],x tB 2 ≥ [uB] def = q(t) = 1 -u - 2 π n∈N e -t 2 (πσ n) 2
n sin(kπn) sin(uπn), k, u ∈ (0, 1). ( 42)

Corollary 4 Let q 0 = lim t→∞ P[d (•) (t) = 0], q u = lim t→∞ P[d (•) (t) ≥ u], u ∈ [1, B + 1],
be the stationary distribution of the number of customers. Then

q 0 = 1 - 1 -λ Eη A B 0 -EA δ+B 0 (EQ δ+B ) -1 , q u = 1 -λ Eη A B+1-u 0 -EA δ+B+1-u 0 (EQ δ+B ) -1 ,
where

ρ i = lim s→0 s -1 ρi (s) = ∞ 0 P[π(t) = i] dt < ∞, Q k = Q 0 k , A u 0 = lim s→0 1 s A u 0 (s) = u i=0 ρ i 1 - Q u-i 1 -λ .
Proof In view of the definition of the process Y k,x (t) and Remark 2, we can write the following equations for the function q s k,x (u), q s 0,x (u) for u ∈ [1, B + 1]

q s k,x (u) = p s k,x (u) + ∞ 0 b s k (x, dy)q s 0,y (u), k ∈ [1, B + 1], (43)
q s 0,y (u) = fy (s) λ B q s B+1,0 (u)

+ (1 -λ) B k=1 λ k-1 q s k,0 (u) ,
where the function b s k (x, dy) = E[e -sb k (x) ; η(x) ∈ dy] is given by (35). Inserting the right-hand side of the second equation into the first, we get

q s k,x (u) = p s k,x (u) + ∞ 0 b s k (x, dy) fy (s)q s B (λ, u),
where q s B (λ, u) = λ B q s B+1,0 (u) + (1λ) B k=1 λ k-1 q s k,0 (u). After some manipulations, we get

q s B (λ, u) = p s B (λ, u) 1 -b(s, λ) -1 , where p s B (λ, u) = λ B p s B+1,0 (u) + (1 -λ) B k=1 λ k-1 p s k,0 (u), b(s, λ) = λ B ∞ 0 b s B+1 (0, dy) fy (s) + (1 -λ) B k=1 λ k-1 ∞ 0 b s k (0, dy) fy (s).
Employing [START_REF] Kadankov | Some characteristics of the one line queuing system with losses[END_REF], (32), (35) and performing necessary calculations, we obtain

∞ 0 b s k (x, dy) fy (s) = 1 -f (s) C s k-1 (x) - Q s k-1 (x) 1 -λ EQ s δ+B EQ s δ+B -Q s B , 1 -b(s, λ) = 1 -f (s) EQ δ+B EQ s δ+B -Q s B , p s B (λ, u) = (1 -λ) A B+1-u 0 (s) -EA δ+B+1-u 0 (s) EQ s δ+B -Q s B , q s B (λ, u) = 1 -λ 1 -f (s) A B+1-u 0 (s) -EA δ+B+1-u 0 (s) EQ s δ+B -1 .
In view of these equalities and the equations of the system (43), we derive the second formula of (41). Taking into account the definition of the process Y k,x (t) and Remark 2, we find for the functions q s k,x (0), q s 0,x (0) that

q s k,x (0) = ∞ 0 b s k (x, dy)q s 0,y (0), ( 44
)
q s 0,y (0) = 1 -fy (s) + fy (s) λ B q s B+1,0 (0) + (1 -λ) B k=1 λ k-1 q s k,0 (0) .
Inserting the right-hand side of the second equation into the first, we get

q s k,x (0) = ∞ 0 b s k (x, dy) 1 -fy (s) + ∞ 0 b s k (x, dy) fy (s)q s B (λ), ( 45
)
where q s B (λ) = λ B q s B+1,0 (0) + (1λ) B k=1 λ k-1 q s k,0 (0). After some transformations of the latter equation, we obtain

q s B (λ) = 1 -1 -b(s, λ) 1 -b(s, λ) -1 , where b(s, λ) = λ B b s B+1 (0) + (1 -λ) B k=1 λ k-1 b s k (0) = 1 -(1 -λ) A B 0 (s) -EA B+δ 0 (s) EQ s δ+B -Q s B .
The latter and the previous equality imply that

q s B (λ) = 1 - 1 -λ 1 -f (s) A B+ 0 (s) -EA δ+B 0 (s) EQ s δ+B -1 .
Inserting the right-hand side of this equality into (45) yields the first equality of (41).

For k, u ∈ (0, 1), denote q t k (x, u, B) = P[d [kB],x (tB 2 ) ≥ [uB]]. Employing (41), the limiting equality [START_REF] Ohsone | Queues with finite waiting room[END_REF], as well as [START_REF] Takagi | Queueing Analysis, A Foundation of Performance Evaluation[END_REF], we find that

1 s lim B→∞ q s/B 2 [kB],x [uB] = lim B→∞ ∞ 0 e -st q t k (x, u, B) dt = 1 -cosh((k -u) + √ 2s/σ ) s + 1 s cosh(k √ 2s/σ ) sinh( √ 2s/σ ) sinh (1 -u) √ 2s/σ def = q * (s), ( 46
)
where u + = max{0, u}. When u ∈ [k, 1), we derive from this formula that

q * (s) = 1 s cosh(k √ 2s/σ ) sinh( √ 2s/σ ) sinh (1 -u) √ 2s/σ , u∈ [k, 1).
It is clear that s 0 = 0 is a simple pole of the function q * (s). On the half-plane (s) < 0, this function has simple poles at s n = -1 2 (σ πn) 2 , n ∈ N, and it is analytic on the entire plane apart from these points. Hence, for α > 0,

q(t) = 1 2πi α+i∞ α-i∞ e st q * (s) ds = n∈Z + Res s=s n q * (s).
Calculating the residues of the function q * (s) in s n , we obtain the right-hand side of Formula (42) for u ∈ [k, 1). One can see that for u ∈ (0, k] the first term on the righthand side of (46) is analytic on the whole plane. Applying the inversion formula, we find that the contour integral of this term is equal to zero. The second term of (46) is the same also for u ∈ [k, 1). Thus, Formula (42) holds for u ∈ (0, 1).

Observe that lim s→0 A u x (s) = lim s→0 EA δ+u 0 (s) = 0, lim s→0 b s k (x) = 1. It follows from (41) and the properties of Laplace transforms that

lim s→0 q s k,x (u) = lim s→0 q s 0,x (u) = q u = lim t→∞ P d (•) (t) ≥ u , u∈ [1, B + 1], lim s→0 q s k,x (0) = lim s→0 q s 0,x (0) = q 0 = lim t→∞ P d (•) (t) = 0 .
Calculating the limits on the right-hand sides of (41) as s → 0 yields the equalities of Corollary 4.

Virtual waiting time

Suppose that at time t 0 = 0 the system is at the state (k, x), k ∈ [0, B + 1], x ≥ 0. Denote by W k,x (t) time required to serve the customers present in the system at time t. Formally, this random variable can be determined in the following way. Let

τ (k) = inf{t : π(t) ≥ k}, k ∈ Z + . Then W k,x (t) = τ d k,x (t) , Ee -pW k,x (t) = B+1 i=0 P d k,x (t) = i Ee -p τ (i) , p >0. Corollary 5 Let k ∈ [0, B + 1], x ≥ 0.
The following equality holds

P W k,x (ν s ) ≤ v = 1 - B i=0 P π(v) = i q s k,x (i + 1), lim t→∞ P W k,x (t) ≤ v = 1 - B i=0 P π(v) = i q i+1 ,
where the distributions q s k,x (u), q u , u ∈ [0, B + 1] are given by (41) and (42).

Proof It is clear that P[ τ (k) > t] = P[π(t) < k]. Hence, Ee -p τ (k) = 1 -P π(ν p ) < k = 1 - k-1 i=0 ρi (p), where ρi (p) = p ∞ 0 e -pv P[π(v) = i] dv. Then Ee -pW k,x (ν s ) = 1 - B+1 i=0 P d k,x (ν s ) = i i-1 j =0 ρj (p) = 1 - B i=0
ρi (p)q s k,x (i + 1).

The right-hand side of this equality implies the formulae of Corollary 5.

G|M κ |1|B system

In this subsection, we consider a partial case of the G δ |M κ |1|B system, namely when P[δ = 1] = 1. This translates to the queueing system where the customers are served one by one. Technically, one has to set the parameter λ = 0 for the geometrical distribution: ) of the random variable δ. In other words, it means that the process {D x (t)} t≥0 has unit negative jumps at the time instants {η n (x)} n∈N and δ N x (t) = N x (t). Then, it follows from (2) that

P[δ = n] = (1 -λ)λ n-1 , n ∈ N, λ ∈ [0, 1
D x (t) = π(t) -N x (t) ∈ Z, t ≥ 0. ( 47
)
We will call this process a difference of a compound Poisson process and a simple renewal process. Setting the parameter λ = 0 in the statements of Lemma 2 leads to the following result. Statements of the lemmas and theorems derived in the previous subsections can be reformulated in a similar way. Letting λ = 0 in the defining Formula (10) for all s, x ≥ 0, we get

Q s k (x) = 1 2πi |z|=α 1 z k+1 fx (s -k(z)) f (s -k(z)) -z dz, α ∈ 0, c(s) , (48)
which is a resolvent sequence of the process {D x (t)} t≥0 defined by (47). This resolvent sequence was introduced in [START_REF] Kadankov | Some characteristics of the one line queuing system with losses[END_REF]. Setting λ = 0 in ( 11) and ( 12), we obtain

f k (x, dl, m, s) = e -s(l-x) 1 -F (l) 1 -F (x) I{l > x}p m k d(l -x) + Φ s 0 (dl, m) Q s k (x) -e -sl 1 -F (l) k i=0 Q s i (x) p m k-i (dl), (49) Ee -sτ k (x) = 1 - s s -k(c(s)) Q s k (x) -A k x (s),
i.e., the Laplace transforms of the upper one-boundary functionals of the process {D x (t)} t≥0 in (47), where ρi

(s) = s ∞ 0 e -st P[π(t) = i] dt, A k x (s) = k i=0 ρi (s) 1 -Q s k-i (x) , Φ s 0 (dl, m) = e -sl 1 -F (l) k∈Z + c(s) k p m k (dl).
We have introduced the auxiliary functions and the resolvent sequence of the process (47); therefore, we can state the following result.

Corollary 6 Let {D x (t)} t≥0 be the difference of the compound Poisson process and the renewal process (see (47)), let {Q s k (x)} k∈Z + , x ≥ 0, be the resolvent sequence of the process given by (48

) with Q s k def = Q s k (0). The Laplace transforms V x r (m, s), V k
x (dl, m, s) of the joint distribution of {χ, L, T } satisfy the following equalities for all x, s ≥ 0, m ∈ N,

V r (x, i, s) = Q s k (x) Q s B+1 δ i1 , V k (x, dl, i, s) = f k (x, dl, i, s) - Q s k (x) Q s B+1 f B+1 (0, dl, i, s),
where δ ij is the Kronecker symbol and f k (x, dl, m, s) is given by (49).

For the Laplace transforms of the first exit time χ from the interval by the process {D x (t)} t≥0 , the following formulae hold

E e -sχ ; A r = Q s k (x) Q s B+1 , E e -sχ ; A k = 1 -A k x (s) + Q s k (x) Q s B+1 1 -A B+1 0 (s) .
In order to prove the corollary, one has to put λ = 0 in the statements of Theorem 1. We now illustrate how the results obtained in Corollaries 1-5 can be applied for studying the queueing system G|M κ |1|B (P[δ = 1] = 1) with a finite buffer.

(x) = 1 -A k-1 x (s) + Q s k-1 (x) A B+1 0 (s) -A B 0 (s) Q s B+1 -Q s B , k ∈ [1, B + 1],
the random variable b k (x) is proper with a finite mathematical expectation In order to invert the Laplace transforms, we have calculated the residues on the right-hand side of (52) at the simple poles p = 0, ± √ 2s/σ. The second part of the first equality [START_REF] Takagi | Queueing Analysis, A Foundation of Performance Evaluation[END_REF] We will now verify the second formula of ( 22). Denote q s k = EQ s δ+k -Q s k , k ∈ Z + . Employing ( 9) and [START_REF] Kadankov | Intersections of an interval by a difference of a Poisson process and a compound renewal process[END_REF], we determine the generating function of this sequence:

Eb k (x) = A k-1 x -Q k-1 (x) A B+1 0 -A B 0 Q B+1 -Q B < ∞.
qs θ = k∈Z + θ k q s k = (1 -λ)(1 -f (s -k(θ ))) (1 -λ) f (s -k(θ)) + λ -θ , θ ∈ 0, c(s) .
This generating function is such that for θ = e -p , p >ln c(s), The third formula of ( 22) can be verified analogously.

= 2 ∞ 0 e

 20 Denote by w {t≥0} the symmetric Wiener process with dispersion σ. Let τ a = inf{t : w t ≥ a} be the first passage time of the level a ∈ R + . The Lévy identity P[τ ≤ t] = 2P[w t ≥ a] implies the following relation for the Laplace transforms -st P[w t ≥ a] dt.

Lemma 7

 7 For s > 0 the equation z = f (sk(z)) has a unique solution c(s) inside the circle |z| < 1. This solution is positive, c(s) ∈ (0, 1). If E[κ], E[η] < ∞, ρ = μE[κ] E[η], then for ρ > 1, lim s→0 c(s) = c ∈ (0, 1); and for ρ ≤ 1, lim s→0 c(s) = 1.

Corollary 7

 7 Let b s k (x) = E[e -sb r (x) ; b r (x) < ∞] be the Laplace transform of the busy period of (k, x) type of the G|M κ |1|B system. Then (i) The following equality holds b s k

(Corollary 8 1 .es/B 2 e

 812 ii) The Laplace transform b s k (x, dy) = E[e -sb k (x) ; η(x) ∈ dy] of the joint distribution of {b k (x), η(x)} is such that for k ∈[1, B + 1] b s k (x, dy) = f k-1 (x, dy, s) -Q s k-1 (x) f B+1 (0, dy, s)f B (0, dy, s) Q s B+1 -Q s B. Let l k (x) be the time of the first loss of a customer in the G|M κ |1|B system, i k,x (ν s ) be the number of the lost customers on the time interval [0, ν s ]. Then(i) The Laplace transform l s k (x) = E[e -sl k (x) ; l k (x) < ∞] of l k (x) is such that l s k (x) = fx (s) + (1 -f (s))S s k-1 (x) f (s) + (1 -f (s))S s B+1 , k ∈ [0, B + 1],whereS s k (x) = k i=0 Q s i (x), S s k (x) = 0 for k < 0. The random variable l k (x) is proper with a finite mathematical expectation El k (x) = Eη x -Eη + Eη[S B+1 -S k-1 (x)] < ∞. (ii) The distribution I s k,x (n) = P[i k,x (ν s ) = n], n ∈ Z + ofthe number of the lost customers on the time interval [0, ν s ] obeys the equalityI s k,x (n) = I {n=0} 1l s k (x) + I {n∈N} l s k (x) 1l s B+1 (0) l s B+1 (0) n-We now derive the asymptotic expansions for the functionS s/B 2 [kB] (x) as B → ∞. The generating function λ) fx (sk(θ)) (1λ) f (sk(θ)) + λθ , θ ∈ 0, c(s) , is such that for θ = e -p , p >ln c(s), {k}p e -pk S s [k] (x) dk = S s e -p (x), p >ln c(s),where {a} is the fractional part of the number a. By S s p (x) = ∞ 0 e -pk S s [k] (x) dk, p >ln c(s), we denote the Laplace transform of the function S s [k] (x). It is clear that S s p (x) ≤ S s e -p (x) ≤ e p S s p (x). (51) Employing the limiting equalities (50) and the definition of the function S s θ (x), we obtain lim B→∞ S -p/B (x)B -3 = lim B→∞ B -3 1e -p/B (1λ) fx (s/B 2k(e -p/B )) (1λ) f (s/B 2k(e -p/B )) + λe -ptransforms (with respect to p) on both sides, we obtain lim

  can be verified analogously:lim B→∞ B -2 ES s/B 2 δ+[kB] = 1 sEη cosh(k √ 2s/σ -1).

e 1 ( 1 - 1 2 p 2 σ 2 1 2 p 2 σ 2

 111212 {k}p e -pk q s [k] dk.By q s p = ∞ 0 e -pk q s [k] dk, p >ln c(s), we denote the Laplace transform of the function q s [k] . It is clear that q s p ≤ qs e -p ≤ e p q s p . (53)In view of the limiting equalities (50) and the definition of the function qs θ , we find limB→∞ qs/B 2 e -p/B B -1 = lim B→∞ (1λ)(1 -f (s/B 2k(e -p/B )))B -λ) f (s/B 2k(e -p/B )) + λe -p/B = pμEκ s .Inverting the Laplace transforms (with respect to p) on both sides, we obtain lim

Corollary 9

The distribution of the number of customers in the system G|M κ |1|B at time ν s is such that

where

be the stationary distribution of the number of customers in the G|M κ |1|B system. Then

where

In order to prove Corollaries 7-9, it suffices to set λ = 0 in the formulae of Theorem 5 and of Corollaries 1-5. The resolvent coefficients Q k , k ∈ Z + , can be obtained form the following recurrence relation

where

Appendix

Let us verify Equalities [START_REF] Takagi | Queueing Analysis, A Foundation of Performance Evaluation[END_REF]