Lamia Lakaour

Djamil Aissani

Karima Adel-Aissanou

Kamel Barkaoui

Sofiane Ziani

An unreliable single server retrial queue with collisions and transmission errors

Keywords: Retrial queue, Collisions, Transmission errors, Unreliable server, Generating function

The present paper deals with the performance evaluation of an M/M/1 retrial queue with collisions, transmission errors and unreliable server. To the best of our knowledge, there are no works that have dealt with retrial queues by considering all the above-mentioned aspects (collisions, transmission errors and unreliable server). This queue can be used as a mathematical model of several computer systems and telecommunication networks. We apply the generating function method to derive the joint distribution of the server state and the orbit length in the steady state, and we obtain some performance measures. Finally, we provide numerical illustrations to show the effectiveness and the applicability of the model.

Introduction

Wireless data networks, which are widely used nowadays, are subject to several problems such as collisions, transmission errors and server breakdowns. These motivate us to devise a stochastic model that takes into account all the aspects causing packets re-transmission to solve such problems. In this study, we consider a retrial queueing model which can be used to analyze the random access to the channel in wireless communications where the server corresponds to the channel radio and the RF module (Radio Frequency module), and the customers are the nodes packets. However, in the distributed coordination function (DCF) mode of IEEE 802.11 basic access scheme, the loss and the re-transmission of packets can be due to either collisions, transmission errors and breakdowns.

In wireless networks, collision takes place when at least two customers access the channel at the same time. In such a situation, the receivers can not decode any frame (packet) correctly. In this paper, we consider that the collision occurs when an arriving customer from outside the system namely a primary customer or a retrial customer (secondary customer) has access directly to the busy server. In this case, both the arriving customer and the customer in service will be moved to the orbit.

Usually, transmission errors are caused by an erroneous packet or by non-ideal channel conditions. Co-channel and adjacent channel interferences may also cause transmission errors (see [START_REF] Yeo | Packet error model for the IEEE 802[END_REF] and [START_REF] Yin | The analysis of performance of IEEE 802.11 MAC protocol using Markov chain[END_REF]). In our model, we consider that the packet is transmitted without transmission errors with probability γ.

The loss and re-transmission of a packet may be also due to the server failure. The failures could be caused by natural disasters, hardware failures, as well as from human actions such as cyber attacks, which are becoming more and more sophisticated. Failures of radio channel are also due to the weakness of signal due to interference or fading, etc.

In the literature, we find works on queues with repeated attempts, some of them take into account collisions and others consider the server failures, but on the best of our knowledge, there is no work which takes into account transmission errors and the other factors together (retrials, collisions and server failures) and this is the object of the paper.

In the proposed model, an arriving customer repeats its attempts to enter the service if 1. the server is busy; 2. there is a transmission error; 3. there is a collision; 4. the server breaks down.

Queues with repeated attempts play a vital role in the performance analysis of many telecommunication processes, including local and wide area networks, switching systems, shared bus local area networks, digital cellular mobile networks, etc. The main feature of a retrial queue is to serve immediately the arriving customer if the server is free. However, if the server is busy, the arriving customer leaves the service area, joins a virtual group of the blocked customers, called orbit, and tries to get service again in a random order and at random time intervals [START_REF] Dragieva | A finite source retrial queue: number of retrials[END_REF]).

Many retrial approaches have been proposed in the literature. The most usual one, described in the classical theory of retrial queues, is the so-called classical retrial policy where each customer (packet, call, etc.) in the orbit tries to get the service independently of the others. In this policy the intervals between successive repeated attempts are exponentially distributed with rate jν, when the orbit size is j (see [START_REF] Aissani | Optimal control of an M/G/1 retrial queue with vacations[END_REF]). Another type of retrial policy, called constant retial policy, is well known for modeling ALOHA protocol. According to this policy, the time between two successive repeated attempts is controlled by an electronic device. Thus, the retrial rate is (1δ j,0 )η, where δ j,0 denotes Kronecker's delta and j the number of repeated customers [START_REF] Fayolle | A simple telephone exchange with delayed feedbacks[END_REF]). [START_REF] Artalejo | Steady-state solution of a single server queue with linear repeated requests[END_REF] have proposed a linear retrial policy. This policy, which is also adopted in the present paper, combines both the last cited approaches (i.e., classical and constant retrial policies). Consequently, its rate is (1δ j,0 )η + jν.

In many situations involving data transmission from diverse sources there can be conflict for a limited number of channels or other facilities. Uncoordinated attempts by several sources to use a single server facility can result in 'Collision' leading to the loss of the transmission. [START_REF] Jonin | Determination of probabilistic characteristic of single-line queues with double connections and repeated calls[END_REF] and [START_REF] Falin | On single-line queues with double connections[END_REF] have analyzed a type of retrial queueing system with collisions, called queue with double connections. For this queue, if an arriving customer finds the service facility busy, interrupts the service of the customer in service, both the arriving customer and the served customer will join the orbit and the server becomes free immediately. In Choi et al. (1992), the authors have analyzed a retrial queueing model with collisions and constant retrial rate arising from unslotted CSMA/CD (Carrier Sense Multiple Access with Collision Detection) protocol. They have derived the generating function of the limiting distribution of the orbit size when the server is idle. The service time of a customer consists of two consecutive phases. The collision occurs when the arriving customer finds the server busy with the first phase of a customer service time. [START_REF] Kim | Retrial queueing system with collision and impatience[END_REF] has considered a Markovian single server retrial queue with collisions and impatience. [START_REF] Kumar | A single server feedback retrial queue with collisions[END_REF] have analyzed the feedback retrial queueing system with collisions and linear retrial policy. [START_REF] Wu | A discrete-time Geo/G/1 retrial queue with preemptive resume and collisions[END_REF] have analyzed a retrial queueing system with preemptive resume and collisions. [START_REF] Kvach | Sojourn Time Analysis of Finite Source Markov Retrial Queuing System with Collision[END_REF] have considered the M/M/1/N retrial queue with collisions, while [START_REF] Jailaxmi | Performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collision[END_REF] have conducted a performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collisions.

In [START_REF] Kim | Retrial queueing system with collision and impatience[END_REF]; [START_REF] Kumar | A single server feedback retrial queue with collisions[END_REF]; [START_REF] Kvach | Sojourn Time Analysis of Finite Source Markov Retrial Queuing System with Collision[END_REF]; [START_REF] Jailaxmi | Performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collision[END_REF], if an arriving customer finds the server busy, the collision occurs between the arriving customer and the customer in service, resulting in the move of both customers to the retrial group. In all of the above mentioned works, the authors consider the collision as the only cause of unsuccessful transmission and retransmission, which is not the case of the real-world communication systems. In this paper, other aspects causing an unsuccessful transmission and re-transmission are considered.

In many real-life situations the server is subject to unpredictable breakdowns. Therefore, queueing systems considering server breakdowns may provide a realistic representation of such situations. This type of systems can be encountered in communication networks, manufacturing and computer systems. Retrial queues dealing with servers failures and repairs have been introduced by [START_REF] Aissani | On the M/G/1 queueing system with repeated orders and unreliable server[END_REF] and [START_REF] Kulkarni | Retrial queues with server subject to breakdowns and repairs[END_REF]. Several other works have been devoted to this field [START_REF] Aissani | Unreliable queueing with repeated orders[END_REF][START_REF] Aissani | A retrial queue with redundancy and unreliable server[END_REF]; Anisimov andAtadzhanov (1992, 1994); [START_REF] Artalejo | New results in retrial queueing systems with breakdown of the servers[END_REF]; [START_REF] Sherman | An M/M/1 retrial queue with unreliable server[END_REF]; [START_REF] Falin | An M/G/1 retrial queue with an unreliable server and general repairs times[END_REF]; [START_REF] Saggou | Performance measures of M/G/1 retrial queues with recurrent customers, breakdowns, and general delays[END_REF]. In [START_REF] Falin | An M/G/1 retrial queue with an unreliable server and general repairs times[END_REF], an M/G/1 retrial queues with an unreliable server and general repair times have been considered. The author has found the joint distribution of the server state, the number of customers in the queue and the number of customers in the orbit in steady state when both service and repair times have a general distribution. [START_REF] Li | A BMAP/G/1 retrial queue with a server subject to breakdowns and repairs[END_REF] have considered a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs. They have assumed that the lifetime of the server is exponentially distributed while the repair time is general. In their work, the supplementary variable has been used. [START_REF] Li | An M/G/1 Retrial Gqueue with general retrial times and working breakdowns[END_REF] have analyzed an M/G/1 retrial Gqueue with working breakdowns. They have supposed that during the breakdown period, the service still continues with a lower rate. In order to discuss the probability generating function of the number of customers in the retrial queue, the supplementary variable technique has been used. [START_REF] Sherman | An M/M/1 retrial queue with unreliable server[END_REF] have analyzed an unreliable M/M/1 retrial queue where the capacity of the orbit and the normal queue are considered infinite. They have provided stability condition as well as several stochastic decomposability results. [START_REF] Wu | A single-server retrial Gqueue with priority and unreliable server under Bernoulli vacation schedul[END_REF] have analyzed an M/G/1 retrial Gqueue with priority under the Bernoulli vacation schedule subject to the server breakdowns and repairs using the matrix geometric method. The retrial Gqueues with server breakdowns have been discussed in many papers such as [START_REF] Do | Bibliography on G-networks, negative customers and applications[END_REF]), Rakhee et al. (2013), [START_REF] Yang | An M [X] /G/1 retrial Gqueue with single vacation subject to the server breakdown and repair[END_REF], [START_REF] Gao | Performance and reliability analysis of an M/G/1 -G retrial queue with orbital search and non-persistent customers[END_REF], [START_REF] Bhagat | N-policy for M x /G/1 unreliable retrial Gqueue with preemptive resume and multi-services[END_REF] and [START_REF] Chen | Batch arrival retrial Gqueue with orbital search and nonpersistent customers[END_REF]. Several detailed reviews of retrial queues can be found in the bibliographies of Artalejo (1999[START_REF] Artalejo | Accessible bibliography on retrial queues: Progress in 2000-2009[END_REF], the survey papers of [START_REF] Yang | A survey on retrial queues[END_REF], [START_REF] Aissani | On the M/G/1 queueing system with repeated orders and unreliable server[END_REF], [START_REF] Falin | A survey of retrial queues[END_REF], [START_REF] Kulkarni | Retrial queues revisited[END_REF] and [START_REF] Kim | A survey of retrial queueing systems[END_REF], the books of [START_REF] Falin | Retrial queues[END_REF] and [START_REF] Artalejo | Retrial queueing systems: a computational approach[END_REF].

As stated above, several studies dealing with retrial queueing systems with collisions and retrial queues with unreliable servers have been considered in the literature. In these studies, the authors consider in their model of retrial queue only one event namely collisions or server breakdowns, which constitute their main draw back. In communication systems, retransmitting packets due to transmission errors caused by an erroneous packet, or non-ideal channel conditions is also one of the frequent factor in retransmitting data and not much work taking this aspect in their model of retrial is found in the literature. However, the consideration of retrial queueing systems taking into account all factors that cause the re-transmission of customers (collisions, transmission errors and server breakdowns) seems interesting. To the best of our knowledge, there are no works considering all these aspects together in the literature. To face these issues, an unreliable server retrial queueing system with linear retrial rate, collisions of customers and transmission errors is investigated in this paper. To obtain some performance measures, the generating function method is applied. To assess our model, we give some numerical examples.

The next section presents the model description. Sect. 3 provides the steady-state probabilities of the orbit size and the server state, and gives the stability condition of the system. In Sect. 4, some performances are obtained using steady-state analysis, while in Sect. 5, numerical examples are given to illustrate the effectiveness of the theoretical results. Finally, the paper is concluded in Sect. 6.

Model description

let us consider an M/M/1 retrial queue with collisions, transmission errors, unreliable server and linear retrial rate. Primary customers arrive from outside the system according to a Poisson process with rate λ. It is assumed that there is no waiting space and an arriving primary customer enters the service with probability p or enters and waits in the orbit of infinite size with probability (1-p) and tries to get served again. The definition of the probability "p" is motivated in modeling random access protocols in wireless communication networks to reduce collisions, i.e, Consider a communication line with slotted time, which is shared by several stations. The duration of the slot equals the transmission time of a single packet of data. If two or more stations are transmitting packets simultaneously, then a collision takes place, i.e. all packets are destroyed and must be retransmitted. If the stations involved in the conflict would try to retransmit destroyed packets in the nearest slot, then a collision occurs with certainty. To avoid this, each station independently of other stations, transmits the packet with probability p and delays actions with probability (1p). Each station introduces a random delay before next attempt to transmit the packet (see [START_REF] Artalejo | Standard and retrial queueing systems: a comparative analysis[END_REF]).

The inter-retrial time of each customer in the orbit is exponentially distributed and independent of the number of customers applying for the service. We assume that the repeated attempts follow the linear retrial policy; the probability of a repeated attempt from the orbit during the interval time (t, t + ∆t) is (η(1δ j,0 ) + jν)∆t + o(∆t), where j is the customers number at time t and δ j,0 denotes Kronecker's delta. The customer in service could either end service successfully (without transmission error) and leaves the system with probability γ (0 < γ ≤ 1) or join the retrial group (due to transmission error) with probability (1γ). If the server is busy, the arriving customer (primary or secondary) collides with the customer in service resulting in both being shifted to the orbit. We consider that the server is subject to random breakdowns only when a customer is in service and this according to a poisson process with rate α. Customers whose service is interrupted by a breakdown have to leave the service zone and join the retrial queue. The repair times are considered exponentially distributed with repair rate β.

We assume that the service times are independent and follow an exponential distribution with mean 1/µ. The stochastic processes involved in the system are considered to be mutually independent.

The system is described by the bivariate process Y(t) = {C(t), X(t); t ≥ 0}, where X(t) denotes the number of customers in the orbit at time t and C(t) denotes the server state defined as

C(t) =          0,
when the server is idle at time t; 1, when the server is busy at time t; 2, when the server breaks down and under repair at time t.

It is clear that the process Y(t) is a continuous time Markov chain, irreducible, aperiodic and time-homogeneous with state space

S = {(0, j), (1, j), (2, j ′ )/j, j ′ ∈ Z + ; j ′ 0}.
3 Analysis of the steady-state distribution (1,1) In order to study the stability of the M/M/1 retrial queue with collisions, transmission errors, unreliable server and linear retrial policy, we consider the embedded Markov chain defined by {X n = X(t n + 0), (n ≥ 0)}, with t 0 = 0, where t n is the time when the server becomes idle for the n th time due to the service completion of the n th customer, collision of customers or reparation of the server, and X n is the number of customers in the orbit just after t n .

Let P j = lim t→∞ P{C(t) = 0, X(t) = j}, Q j = lim t→∞ P{C(t) = 1, X(t) = j}, j ≥ 0 and R j = lim t→∞ P{C(t) = 2, X(t) = j}, j ≥ 1,
(1,2) (1,j) (1, j+1) … … β β β β (1-p) λ (1-p) λ (1-p) λ (1-p) λ (1-p) λ (1-p) λ λ p ϒ μ ϒ μ (1-p) λ (1-p) λ (1-p) λ (1-p) λ (1-p) λ (1-p) λ ϒ μ ϒ μ λ p λ p λ p (1 -ϒ ) μ ( + ) ( + 2 ) ( + 3 ) ( + j ) ( + (j + 1 ) ) ( + ( j+ 2 ) ) (1 -ϒ ) μ + (η + ѵ ) (1 -ϒ ) μ + (η + 2 ѵ ) (1 -ϒ ) μ + (η + (j -1 )ѵ ) ( 1 -ϒ ) μ + ( η + ( j+ 1 ) ѵ ) (1 -ϒ ) μ + (η + jѵ ) λ λ λ λ λ (2,1) (2,2) (2,j) (2,

Proposition 1

The embedded Markov chain (X n ) n≥0 is ergodic if and only if

ρ = λ(2β + α) βγµ < 1, (1)
where the parameters λ, β, α and γ are as defined above.

Proof 1

To prove the sufficient condition, we use the criterion based on the theory of mean drift. The most important result of this theory is the Foster's criterion (see [START_REF] Pakes | Some conditions for ergodicity and recurrence of Markov chains[END_REF] and [START_REF] Tweedie | Sufficient conditions for regularity, recurrence and ergodicity of Markov processes[END_REF]).

In our model the limiting mean drift is lim

k→∞ △ k = E[X n+1 -X n | X n = k] = λ(2β + α) -βγµ, ∀k > j 0 , j 0 ≥ 0. (X n ) n≥0 is ergodic if lim k→∞ △ k < 0. Hence λ(2β + α) < βγµ if and only if λ(2β+α) βγµ = ρ < 1.
It can also be verified that this condition is necessary for ergodicity.

Using the balance equations of the Markovian system illustrated in Fig. 1, we obtain

λP 0 = γµQ 0 , (2)
(λ + η + ν)P 1 = (1 -p)λP 0 + (1 -γ)µQ 0 + γµQ 1 + βR 1 , (3)
(λ + η + jν)P j = (1 -p)λP j-1 + (η + (j -1)ν)Q j-1 + (1 -γ)µQ j-1 + λpQ j-2 + βR j + γµQ j , j = 2, 3, ..., (4)
(α + λ + µ)Q 0 = λpP 0 + (η + ν)P 1 , (5)
(α + λ + µ + η + jν)Q j = λpP j + (η + ( j + 1)ν)P j+1 + (1 -p)λQ j-1 , j = 1, 2, 3, ..., (6)
(λ + β)R 1 = αQ 0 , (7)
(λ + β)R j = αQ j-1 + λR j-1 , j = 2, 3, .... (8)
To solve Eqs. ( 2)-( 8), we use the following partial generating functions:

P(z) = ∞ ∑ j=0 P j z j , Q(z) = ∞ ∑ j=0 Q j z j and R(z) = ∞ ∑ j=1 R j z j , z ∈ [0, 1], (9)
where P(z)

( respectively Q(z), R(z)
)

is the PGF (Probability Generating Function) of the number of customers in the orbit when the server is idle (respectively busy, failing and under repair).

By multiplying both sides of Eqs. ( 2)-( 8) by z j and then making a summation, we obtain the following equations:

νzP ′ (z) + (λ + η -(1 -p)λz)P(z) -ηP 0 = (γµ + (1 -γ)µz + λpz 2 + ηz)Q(z) + νz 2 Q ′ (z) -ηzQ 0 + βR(z), (10)
νP ′ (z) + (λp + η z )P(z) - η z P 0 = -λ(1 -p)zQ(z) + νzQ ′ (z) -ηQ 0 + (λ + α + µ + η)Q(z), (11)
(λ(1 -z) + β)R(z) = αzQ(z). ( 12
)
Multiplying both sides of Eq. ( 11) by z, then subtracting the obtained equation from Eq. ( 10) and finally, summing with Eq. ( 12), we obtain

λP(z) = (-λz + γµ)Q(z) -λR(z). ( 13
)
The following theorem gives the different probabilities of the server's state.

Theorem 1

For an M/M/1 retrial queue with collisions, transmission errors, unreliable server and linear retrial policy, under the stability condition λ(2β + α) < βγµ, we have the following probabilities of the server state:

(i) The probability that the sever is busy is

Q(1) = λ γµ . (14)
(ii) The probability that the server breaks down is

R(1) = αλ βγµ . (15)
(iii) The probability that the server is idle is

P(1) = 1 - λ γµ - αλ βγµ . ( 16
)
Proof 2 (i) For z = 1, Eq. ( 13) becomes

λ (P(1) + Q(1) + R(1)) = γµQ(1).
Under the stability condition (1), the limiting probability distribution exists, i.e.

P(1) + Q(1) + R(1) = 1,
then, we obtain the probability that the server is busy given in Eq.( 14).

(ii) From Eq. ( 12), for z = 1 and replacing Q(1) obtained previously, we get the probability that the server breaks down given in Eq.( 15).

(iii) To obtain P(1), we first put z = 1 in Eq. ( 13) and then, we replace Q(1) and R(1) obtained previously.

Now, let us consider the probability generating functions of the number of customers in the orbit when the server is busy, breaks down and under repair, and idle; and the probability generating functions of the number of customers in the orbit and in the system, which are given in the following theorem.

Theorem 2

For an unreliable Markovian single server retrial queue with collisions, transmission errors and linear retrial policy, under the stability condition ρ < 1, we have:

(i) the probability generating function Q(z) of the orbit size when the server is busy is

Q(z) = z -η ν × [ λ γµ exp ( (z 6 -1)A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -1)B 5(γµν(β + λ) 2 ) 6 ) exp ( (1 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (1 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (1 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (1 -z)F (γµν(β + λ) 2 ) 2 ) - λη νγ P 0 ∫ 1 z exp ( (z 6 -u 6 )A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -u 5 )B 5(γµν(β + λ) 2 ) 6 ) exp ( (u 4 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (u 3 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (u 2 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (u -z)F (γµν(β + λ) 2 ) 2 ) u η ν -1 × (γ -λ µ u)(β + λ -λu) -(2λ 2 u -2λ 2 u 2 -βγµ + αλu + 2βλu -γλµ + γλµu) du        ; (17)
(ii) the probability generating function R(z) of the orbit size when the server breaks down and under repair is

R(z) = αz λ(1 -z) + β z -η ν × [ λ γµ exp ( (z 6 -1)A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -1)B 5(γµν(β + λ) 2 ) 6 ) exp ( (1 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (1 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (1 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (1 -z)F (γµν(β + λ) 2 ) 2 ) - λη νγ P 0 ∫ 1 z exp ( (z 6 -u 6 )A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -u 5 )B 5(γµν(β + λ) 2 ) 6 ) exp ( (u 4 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (u 3 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (u 2 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (u -z)F (γµν(β + λ) 2 ) 2 ) × u η ν -1 (γ -λ µ u)(β + λ -λu) -(2λ 2 u -2λ 2 u 2 -βγµ + αλu + 2βλu -γλµ + γλµu) du        ; (18)
(iii) the probability generating function P(z) of the orbit size when the server is idle is

P(z) = ( -z + γµ λ - αz λ(1 -z) + β ) z -η ν [ λ γµ exp ( (z 6 -1)A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -1)B 5(γµν(β + λ) 2 ) 6 ) × exp ( (1 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) exp ( (1 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (1 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (1 -z)F (γµν(β + λ) 2 ) 2 ) - λη νγ P 0 ∫ 1 z exp ( (z 6 -u 6 )A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -u 5 )B 5(γµν(β + λ) 2 ) 6 ) exp ( (u 4 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (u 3 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (u 2 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (u -z)F (γµν(β + λ) 2 ) 2 ) × u η ν -1 (γ -λ µ u)(β + λ -λu) -(2λ 2 u -2λ 2 u 2 -βγµ + αλu + 2βλu -γλµ + γλµu) du        ; (19)
(iv) the probability generating function K(z) for the number of customers in the orbit is

K(z) = (-z + γµ λ + 1)z -η ν [ λ γµ exp ( (z 6 -1)A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -1)B 5(γµν(β + λ) 2 ) 6 ) exp ( (1 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (1 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (1 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (1 -z)F (γµν(β + λ) 2 ) 2 ) - λη νγ P 0 ∫ 1 z exp ( (z 6 -u 6 )A 6(γµν(β + λ) 2 ) 7 ) × exp ( (z 5 -u 5 )B 5(γµν(β + λ) 2 ) 6 ) exp ( (u 4 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) exp ( (u 3 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (u 2 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) × exp ( (u -z)F (γµν(β + λ) 2 ) 2 ) u η ν -1 (γ -λ µ u)(β + λ -λu) -(2λ 2 u -2λ 2 u 2 -βγµ + αλu + 2βλu -γλµ + γλµu) du        ; (20)
(v) the probability generating function H(z) for the number of customers in the system is

H(z) = γµ λ z -η ν [ λ γµ exp ( (z 6 -1)A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -1)B 5(γµν(β + λ) 2 ) 6 ) exp ( (1 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (1 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp ( (1 -z 2 )E 2(γµν(β + λ) 2 ) 3 ) exp ( (1 -z)F (γµν(β + λ) 2 ) 2 ) - λη νγ P 0 ∫ 1 z exp ( (z 6 -u 6 )A 6(γµν(β + λ) 2 ) 7 ) exp ( (z 5 -u 5 )B 5(γµν(β + λ) 2 ) 6 ) exp ( (u 4 -z 4 )C 4(γµν(β + λ) 2 ) 5 ) × exp ( (u 3 -z 3 )D 3(γµν(β + λ) 2 ) 4 ) exp( (u 2 -z 2 )E 2(γµν(β + λ) 2 ) 3 exp ( (u -z)F (γµν(β + λ) 2 ) 2 ) × u η ν -1 (γ -λ µ u)(β + λ -λu) -(2λ 2 u -2λ 2 u 2 -βγµ + αλu + 2βλu -γλµ + γλµu) du        ; (21)
where 4η+β(9-10p)+µ+γµ-2(6λ(-1+p)+γpµ)+ν)-4b(4η+λ(8-12p)+(2+γ-4γp)µ+2ν))+(α+2(β+λ+γµ)) 3 (4γηµ(α+4(β+λ)+γµ)+(α+2(β+λ+γµ))(αη+2γηµ+(β+λ)(α+2η+ µ-γpµ)+αν+βν+λ(β+λ+ν)))-γµ(α+2(β+λ+γµ))(3γηµ(α+4(β+λ)+γµ) 2 +(α+2(β+λ+γµ)) 2 (-αβ+β 2 +3λ 2 + (α+4λ)η+4β(λ+η)-(α+4β)λp-2(β 2 +λ 2 )p+γηµ+2(β+λ)(α+µ-γpµ+ν))+3(α+2(β+λ+γµ))(2γ(β+λ)ηµ+(α+ 4(β+λ)+γµ)(αη+2γηµ+(β+λ)(α+2η+µ-γpµ)+αν+βν+λ(β+λ+ν))))+γ 2 µ 2 (4γ(β+λ)ηµ(α+4(β+λ)+γµ)+(β+ λ)(α+2(β+λ+γµ)) 2 (α+2β+3λ+2η-αp+µ-p(4(β+λ)+γµ)+ν)+(α+4(β+λ)+γµ) 2 (αη+2γηµ+(β+λ

P 0 = ν µη exp ( -A 6(γµν(β + λ) 2 ) 7 ) exp ( -B 5(γµν(β + λ) 2 ) 6 ) exp ( C 4(γµν(β + λ) 2 ) 5 ) exp ( D 3(γµν(β + λ) 2 ) 4 ) × exp ( E 2(γµν(β + λ) 2 ) 3 ) exp ( F (γµν(β + λ) 2 ) 2 ) { lim z→0 + ∫ 1 z exp ( -u 6 A 6(γµν(β + λ) 2 ) 7 ) exp ( -u 5 B 5(γµν(β + λ) 2 ) 6 ) × exp ( u 4 C 4(γµν(β + λ) 2 ) 5 ) exp ( u 3 D 3(γµν(β + λ) 2 ) 4 ) exp ( u 2 E 2(γµν(β + λ) 2 ) 3 ) exp ( uF (γµν(β + λ) 2 ) 2 ) × u η ν -1 (γ -λ µ u)(β + λ -λu) -(2λ 2 u -2λ 2 u 2 -βγµ + αλu + 2βλu -γλµ + γλµu) du        -1 ; (22) A = γλ 2 (β + λ) 4 µν 2 (-λ 4 (β + λ) 4 η(α + 2(β + λ + γµ)) 6 ν 4 -γ 4 λ 4 (β + λ) 6 µ 4 (-8β -10λ -4η + α(-3 + 4p) -(2 + γ)µ + 4p(4(β + λ) + γµ) -2ν)ν 4 + 5(α + 2(β + λ + γµ))(αη + 2γηµ + (β + λ)(α + 2η + µ -γpµ) + αν + βν + γλ 4 (β + λ) 4 ηµ(α + 4(β + λ) + γµ)(α + 2(β + λ + γµ)) 3 ν 4 + λ(β + λ + ν)) -γλ 4 (β + λ) 4 µ(α + 2(β + λ + γµ)) 2 ν 4 (6γηµ(α + 4(β + λ) + γµ) 2 + (α + 2(β + λ + γµ)) 2 (-αβ + β 2 + 3λ 2 + (α + 4λ)η + 4β(λ + η) -(α + 4β)λp -2(β 2 + λ 2 )p + γηµ + 2(β + λ)(α + µ -γpµ + ν)) + 4(α + 2(β + λ + γµ))(2γ(β + λ)ηµ + (α + 4(β + λ) + γµ)(αη + 2γηµ + (β + λ)(α + 2η + µ -γpµ) + αν + βν + λ(β + λ + ν)))) - γ 3 λ 4 (β + λ) 4 µ 3 ν 4 (4γ(β + λ) 2 ηµ -(β + λ) 2 (-1 + 2p)(α + 2(β + λ + γµ)) 2 + 4(β + λ)(α + 4(β + λ) + γµ)(αη + 2γηµ + (β + λ)(α + 2η + µ -γpµ) + αν + βν + λ(β + λ + ν)) + (α + 4(β + λ) + γµ) 2 (-αβ + β 2 + 3λ 2 + (α + 4λ)η + 4β(λ + η) -(α + 4β)λp - 2(β 2 + λ 2 )p + γηµ + 2(β + λ)(α + µ -γpµ + ν)) + 2(β + λ)(α + 2(β + λ + γµ))((α + 4(β + λ) + γµ)(α + 2β + 3λ + 2η -αp + µ - p(4(β + λ) + γµ) + ν) + 2(-αβ + β 2 + 3λ 2 + (α + 4λ)η + 4β(λ + η) -(α + 4β)λp -2(β 2 + λ 2 )p + γηµ + 2(β + λ)(α + µ -γpµ + ν)))) + γ 2 λ 4 (β + λ) 4 µ 2 ν 4 (γηµ(α + 4(β + λ) + γµ) 3 + (β + λ)(α + 2(β + λ + γµ)) 3 (α + 2β + 3λ + 2η -αp + µ -p(4(β + λ) + γµ)+ν)+3(α+4(β+λ)+γµ)(α+2(β+λ+γµ))(4γ(β+λ)ηµ+(α+4(β+λ)+γµ)(αη+2γηµ+(β+λ)(α+2η+µ-γpµ)+ αν + βν + λ(β + λ + ν))) + 3(α + 2(β + λ + γµ)) 2 (2(β + λ)(αη + 2γηµ + (β + λ)(α + 2η + µ -γpµ) + αν + βν + λ(β + λ + ν)) + (α + 4(β + λ) + γµ)(-αβ + β 2 + 3λ 2 + (α + 4λ)η + 4β(λ + η) -(α + 4β)λp -2(β 2 + λ 2 )p + γηµ + 2(β + λ)(α + µ -γpµ + ν))))); B = γλ 5 (β + λ) 7 µν 5 (-η(α + 2(β + λ + γµ)) 5 + γ 3 (β + λ)µ 3 (-20λ 2 -16λη + α 2 (-1 + p) + 24λ 2 p + 12β 2 (-1 + 2p) -8λµ - 5γλµ-4γηµ+16γλpµ-γµ 2 +γ 2 pµ 2 -8λν-γµν-α(
)(α+2η+µ- γpµ)+αν+βν+λ(β+λ+ν))+2(α+2(β+λ+γµ))(2(β+λ)(αη+2γηµ+(β+λ)(α+2η+µ-γpµ)+αν+βν+λ(β+λ+ν))+ (α + 4(β + λ) + γµ)(-αβ + β 2 + 3λ 2 + (α + 4λ)η + 4β(λ + η) -(α + 4β)λp -2(β 2 + λ 2 )p + γηµ + 2(β + λ)(α + µ -γpµ + ν))))); C = -γλ 4 (β + λ) 6 µν 4 (α 4 (β + λ + ν) + 4(β + λ) 4 (2λ -(-2 + γ)µ + 2ν) + α 3 (6β 2 + 7λ 2 + 4γµν + λ(µ + 2γµ + 7ν) + β(13λ + µ -γ(-3 + p)µ + 7ν)) + α 2 (12β 3 + 18λ 3 + 6γ 2 µ 2 ν + γλµ((2 + γ)µ + 14ν) + λ 2 (6µ + 5γµ + 18ν) + β 2 (42λ + (6 + 7γ -4γp)µ + 18ν) + β(48λ 2 + 4λ((3 + 3γ -γp)µ + 9ν) + γµ((2 + 4γ -3γp)µ + 14ν))) + α(8β 4 + 20λ 4 + 4γ 3 µ 3 ν + 4γλ 2 µ(µ + 3ν) + 4λ 3 (3µ + 5ν) + 4β 3 (11λ + 3µ -γpµ + 5ν) + γ 2 λµ 2 (µ + 7ν) + β 2 (84λ 2 + γµ((4 + 3γ -4γp)µ + 12ν)+λ(36µ-8γpµ+60ν))+β(68λ 3 +γ 2 µ 2 (µ-3γ(-1+p)µ+7ν)+γλµ((8+3γ-4γp)µ+24ν)+λ 2 (-4(-9+γp)µ+60ν)))); D = -γλ 3 (β + λ) 5 µν 3 (α 3 (β + λ + ν) + 2(β + λ) 3 (2λ -(-2 + γ)µ + 2ν) + α 2 (4β 2 + 5λ 2 + 3γµν + λ(µ + γµ + 5ν) + β(9λ + µ -γ(-2 + p)µ + 5ν)) + α(4β 3 + 8λ 3 + 3γ 2 µ 2 ν + 4λ 2 (µ + 2ν) + γλµ(µ + 5ν) + β 2 (16λ + (4 + γ -2γp)µ + 8ν) + β(20λ 2 + λ(8 + γ -2γp)µ + 16λν + γµ(µ -2γ(-1 + p)µ + 5ν)))); E = γλ 2 (β+λ) 4 µν 2 (-α 2 (β+λ+ν)-(β+λ) 2 (2λ+2µ-γµ+2ν)-α(2β 2 +3λ 2 +λµ+3λν+2γµν+β(5λ+µ+γµ-γpµ+3ν)));
and

F = γλ(β + λ) 3 µν(-α(β + λ + ν) -(β + λ)(λ + µ -γpµ + ν)).
Proof 3 (i) By differentiating Eq. ( 13) with respect to z and substituting into Eq. ( 10) and using Eq. ( 12), we obtain

Q ′ (z) + 1 ω [ λ 4 (1 -2p)z 4 + λ 3 (α(p -1) -µ + 4pβ + 4pλ -3λ -2β + pγµ -2η -ν)z 3 + λ 2 ((β + λ)(2α -2γpµ + 2(µ + ν)) + 4β(λ + η) + η(α + 4λ) -2p(β 2 + λ 2 ) + β 2 + 3λ 2 -αβp -λp(α + 4β) + ηγµ)z 2 -λ(β + λ)((β + λ)(α + 2η -γpµ + µ) + λ(λ + β + ν) + βν + ηα + αν + 2γηµ)z + γηµ(β + λ) 2 ] Q(z) = λη(γ -λ µ z)(β + λ -λz) 2 γω P 0 , (23)
where ω = -νz

( β + λ(1 -z) ) ( (1 -z)(2λ 2 z -γλµ) -βγµ + λz(α + 2β)
) .

The equation given in Eq.( 23) is a first order differential equation with second member. It has a singular point at z = 0, so the domain of solution is (0,1].

Eq.( 23) has the same form as the one below:

Q ′ (z) + az 4 + bz 3 + cz 2 + dz + e lz 4 + mz 3 + nz 2 + rz Q(z) = g(z), z ∈ (0, 1], (24)
where We used the power series of T(x) in the neighborhood of 0 to get ∫ T(x)dx. Then, we have

∫ T(x)dx = e r log(|x|) - x 6 6r 7 ( (am + bl)r 5 + (-an 2 + (-2bm -2cl)n -cm 2 -2dlm -el 2 )r 4 +(bn 3 + (3cm + 3dl)n 2 + (3dm 2 + 6elm)n + em 3 )r 3 + (-cn 4 + (-4dm -4el)n 3 -6em 2 n 2 )r 2 +(dn 5 + 5emn 4 )r -en 6 ) - x 5 5r 6 ( (an + bm + cl)r 4 + (-bn 2 + (-2cm -2dl)n -dm 2 -2elm)r 3 + ( cn 3 + (3dm + 3el)n 2 + 3em 2 n)r 2 + (-dn 4 -4emn 3 )r + en 5 ) + x 4 4r 5 ( ar 4 + (-bn -cm -dl)r 3 +(cn 2 + (2dm + 2el)n + em 2 )r 2 + (-dn 3 -3emn 2 )r + en 4 ) + x 3 3r 4 (br 3 + (-cn -dm -el)r 2 +(dn 2 + 2emn)r -en 3 ) + x 2 2r 3 ( cr 2 + (-dn -em)r + en 2 ) + x r 2 (dr -en) + o(x 6 ) + constant. ( 29
)
We rewrite this integral as follows ∫ T(x)dx = e r log(|x|) -x 6 6r 7 A -

x 5 5r 6 B + x 4 4r 5 C + x 3 3r 4 D + x 2 2r 3 E + x r 2 F + constant, (30)
where

A = (am + bl)r 5 + (-an 2 + (-2bm -2cl)n -cm 2 -2dlm -el 2 )r 4 + (bn 3 + (3cm + 3dl)n 2 + (3dm 2 + 6elm)n + em 3 )r 3 + (-cn 4 + (-4dm -4el)n 3 -6em 2 n 2 )r 2 + (dn 5 + 5emn 4 )r -en 6 ; B = (an + bm + cl)r 4 + (-bn 2 + (-2cm -2dl)n -dm 2 -2elm)r 3 + (cn 3 + (3dm + 3el)n 2 + 3em 2 n)r 2 + (-dn 4 -4emn 3 )r + en 5 ; C = ar 4 + (-bn -cm -dl)r 3 + (cn 2 + (2dm + 2el)n + em 2 )r 2 + (-dn 3 -3emn 2 )r + en 4 ; D = br 3 + (-cn -dm -el)r 2 + (dn 2 + 2emn)r -en 3 ; E = cr 2 + (-dn -em)r + en 2 ;
and

F = dr -en.
and we have exp

( - ∫ z 1 T(x)dx ) = z -e r exp( z 6 -1 6r 7 A) exp( z 5 -1 5r 6 B) exp( 1 -z 4 4r 5 C) exp( 1 -z 3 3r 4 D) (31) × exp( 1 -z 2 2r 3 E) exp( 1 -z r 2 F); exp (∫ u 1 T(x)dx ) = u e r exp( 1 -u 6 6r 7 A) exp( 1 -u 5 5r 6 B) exp( u 4 -1 4r 5 C) exp( u 3 -1 3r 4 D) (32) × exp( u 2 -1 2r 3 E) exp( u -1 r 2 F).
Substituting the equation corresponding to exp

( - ∫ z 1 T(x)dx ) , exp (∫ u 1 T(x)dx
) and g(u) in Eq.( 28), and after simplifications, we get the probability generating function Q(z) of the orbit size when the server is busy.

(ii) By substituting Eq.( 17), obtained in (i), in Eq.( 12), we obtain the probability generating function R(z) of the orbit size when the server breaks down and under repair.

(iii) By substituting Eq.( 17) and Eq.( 18), obtained in (i) and (ii) respectively, into (13), we obtain the probability generating function P(z) of the orbit size when the server is idle.

(iv) The probability generating function for the number of customers in the orbit denoted by K(z) is defined as K(z) = P(z) + Q(z) + R(z). Substituting expressions for P(z), Q(z) and R(z), we get K(z) given in Eq. ( 20).

(v) The probability generating function for the number of customers in the system, defined as H(z) = P(z) + R(z) + zQ(z), is given in Eq. ( 21).

It can be noted that, as z → 0 + , Q(0) = Q 0 < ∞ and z -η ν diverges. Thus, from (17), we obtain the steadystate joint probability P 0 that the server is idle and no customers are in the orbit.

System characteristics

The following theorem summarizes some important characteristics of an unreliable single server retrial queue with collisions, transmission errors and linear retial rate.

Theorem 3

For a Markovian single server retrial queue with collisions, transmission errors, unreliable server and linear retial rate, under the stability condition λ(2β+α) < βγµ, we have the following performances:

(i) The mean number of customers L q in the orbit is

L q = - λ γµ - η ν + A (γµν(β + λ) 2 ) 7 + B (γµν(β + λ) 2 ) 6 - C (γµν(β + λ) 2 ) 5 - D (γµν(β + λ) 2 ) 4 - E (γµν(β + λ) 2 ) 3 - F (γµν(β + λ) 2 ) 2 - µηβ ν ( γ -λ µ βγµ -αλ -2λβ )P 0 . (33)
(ii) The mean number of customers L s in the system is

L s = - η ν + A (γµν(β + λ) 2 ) 7 + B (γµν(β + λ) 2 ) 6 - C (γµν(β + λ) 2 ) 5 - D (γµν(β + λ) 2 ) 4 - E (γµν(β + λ) 2 ) 3 - F (γµν(β + λ) 2 ) 2 - µηβ ν ( γ -λ µ βγµ -αλ -2λβ )P 0 . (34)
(iii) The mean waiting time W q in the orbit is

W q = - 1 γµ - η λν + A λ(γµν(β + λ) 2 ) 7 + B λ(γµν(β + λ) 2 ) 6 - C λ(γµν(β + λ) 2 ) 5 - D λ(γµν(β + λ) 2 ) 4 - E λ(γµν(β + λ) 2 ) 3 - F λ(γµν(β + λ) 2 ) 2 - µηβ λν ( γ -λ µ βγµ -αλ -2λβ )P 0 . (35)
(iv) The mean waiting time W s in the system is

W s = - η λν + A λ(γµν(β + λ) 2 ) 7 + B λ(γµν(β + λ) 2 ) 6 - C λ(γµν(β + λ) 2 ) 5 - D λ(γµν(β + λ) 2 ) 4 - E λ(γµν(β + λ) 2 ) 3 - F λ(γµν(β + λ) 2 ) 2 - µηβ λν ( γ -λ µ βγµ -αλ -2λβ )P 0 . (36)
(v) The probability R of having an empty system is

R = ν(λ + γµ) µ 2 ηγ exp ( -A 6(γµν(β + λ) 2 ) 7 ) exp ( -B 5(γµν(β + λ) 2 ) 6 ) exp ( C 4(γµν(β + λ) 2 ) 5 ) exp ( D 3(γµν(β + λ) 2 ) 4 ) × exp ( E 2(γµν(β + λ) 2 ) 3 ) exp ( F (γµν(β + λ) 2 ) 2 ) { lim z→0 + ∫ 1 z exp ( -u 6 A 6(γµν(β + λ) 2 ) 7 ) exp ( -u 5 B 5(γµν(β + λ) 2 ) 6 ) × exp ( u 4 C 4(γµν(β + λ) 2 ) 5 ) exp ( u 3 D 3(γµν(β + λ) 2 ) 4 ) exp ( u 2 E 2(γµν(β + λ) 2 ) 3 ) exp ( uF (γµν(β + λ) 2 ) 2 ) × u η ν -1 (γ -λ µ u)(β + λ -λu) -(2λ 2 u -2λ 2 u 2 -βγµ + αλu + 2βλu -γλµ + γλµu) du        -1 . (37)
(vi) The interrupted frequency N of the service caused by collisions is

N = (λp + η) λ γµ - νλ γµY [ λ 4 (1 -2p) + λ 3 (α(p -1) -µ + 4pβ + 4pλ -3λ -2β + pγµ -2η -ν) + λ 2 ((β + λ)(2α -2γpµ + 2(µ + ν)) + 4β(λ + η) + η(α + 4λ) -2p(β 2 + λ 2 ) + β 2 + 3λ 2 -αβp -λp(α + 4β) + ηγµ) -λ(β + λ)((β + λ)(α + 2η -γpµ + µ) + λ(λ + β + ν) + βν + ηα + αν + 2γηµ) + γηµ(β + λ) 2 ] + νβ 2 λη(γ -λ µ ) γY P 0 , (38)
where Y = -νβ(-βγµ + λα + 2λβ).

Numerical illustrations

In this section, we give some numerical illustrations to show the effect of the variation and the impact of some parameters on the system performances.

In communication systems, it is interesting to determine the proportion of time that the channel is free and no packets need to be re-transmitted, i.e., no station ask for transmitting through this channel. Then, we first consider the variation of the steady state joint probability P 0 , that the channel is idle and no packets in the orbit, according to different system parameters. In Figs. 234567, we present the variation of the probability P 0 against the retrial rate ν, for three values of γ (γ = 0.3, γ = 0.5 and γ = 0.7), where γ represents the probability that the packet is transmitted without error. Its values correspond to the channel conditions and packet state (erroneous packet or not).

As it is shown, the probability P 0 increases when the retrial rate ν increases for any value of γ. However, P 0 increases considerably as the probability γ increases. When the probability γ becomes small, the number of packets that need re-transmission due to transmission errors become more important. This generates a large mass of packets in the orbit and consequently a low probability of having an empty orbit.

Really, the retrial rate is based on the stations number that try to transmit through the same channel. Then, it is very important to see the evolution of P 0 for different retrial rate η. Figs. 234show the influence of η on the evolution of P 0 . We notice that for a high retrial rate (η = 5), values of P 0 are more important, this can be explained by the fact that the higher rate η gives more chance to packets from the orbit to be transmitted.

In queueing systems with failures, the repair rate is one of the most important parameters. In telecommunication networks, it can interpret the ability of the system to resume service after an interruption. In Figs. 567, we show the impact of the repair rate β on P 0 , while the other parameters are fixed. From these figures, we notice, as expected, that P 0 increases with increasing the repair rate β, for any value of γ. Recent communication protocols try to minimize collisions but cannot avoid them completely. Calculating the number of service interruptions due to collisions is very beneficial for evaluating these protocols. Then, we illustrate in Fig. 8 how the steady-state interrupted frequency N of the service caused by collisions varies when the probability p increases, for two values of the arrival rate λ (λ = 0.05 and λ = 0.2). We observe that N increases according to the probability p. This effect is less visible when the arrival rate λ is small. Indeed, there is not a great influence on the frequency N because the service rate is high (µ = 2) compared to the arrival rate λ (λ = 0.05). These conditions allow the channel to serve packets without having a lot of arrivals from outside the system and reduce considerably the collisions.

As shown in Figs. 9-11, the number of packets needing re-transmission (L q ) is reduced when the retrial rate become large, this fact is also visible when increasing the probability p. Figs. 12-14, represent the variation of the mean busy period E(L) of the system according to ρ (the utilization factor) for three values of p (p=0, p=0.5, p=1). We observe that E(L) increases according to ρ, this characteristic is reduced when primary packets join directly the channel (p = 1) compared with the case where packets are transmitted with a probability p = 0 or p = 0.5. This appears significantly in the case when the system approaches the saturation (ρ ≃ 1).

Really, this is explained by the fact that when p = 1, packets have more chance to be transmitted without passing through the orbit and thus reduce the number of retrial packets.

Conclusion

In this paper, we have investigated an unreliable Markovian single server retrial queue with collisions, transmission errors and linear retrial policy. This queueing model can be used for performance evaluation of channel random access in communication systems. To that end, we have used the Markov process theory to derive the steady-state equations. We have applied the generating function method to obtain the joint steady-state probability generating functions of the server state and the orbit length, and we have also computed essential characteristics. Finally, we have presented numerical illustrations for several system characteristics based on given numerical values for the system parameters. The obtained numerical results confirm the effectiveness of our analytical research. Several future works could be drawn from our results. First, It is possible to work on some system parameters like (µ, p, β) to obtain a desired value of P 0 . Then, one can develop an adaptive algorithm for the management of the probability p to according to a fixed E(L) value. It is also It is also interesting to consider our result to emerge to cognitive radio management.

  be the joint distributions of the server state and the number of customers in the orbit in the steady state.

Figure 1 :

 1 Figure 1: Transition diagram of an M/M/1 retrial queue with collisions, transmission errors, unreliable server and linear retrial policy.

Figure 2 :

 2 Figure 2: P 0 as function of ν for λ = 0.05, η = 1.5, µ = 2, α = 10, β = 10, p = 0.

Figure 3 :

 3 Figure 3: P 0 as function of ν for λ = 0.05, η = 3, µ = 2, α = 10, β = 10, p = 0.

Figure 4 :

 4 Figure 4: P 0 as function of ν for λ = 0.05, η = 5, µ = 2, α = 10, β = 10, p = 0.

Figure 5 :

 5 Figure 5: P 0 as function of ν for λ = 0.05, p = 0, µ = 2, α = 5, β = 3, η = 2.

Figure 6 :

 6 Figure 6: P 0 as function of ν for λ = 0.05, p = 0, µ = 2, α = 5, β = 4, η = 2.

Figure 7 :

 7 Figure 7: P 0 as function of ν for λ = 0.05, p = 0, µ = 2, α = 5, β = 5, η = 2.

Figure 8 :

 8 Figure 8: N versus p for γ = 0, 5, η = 3, µ = 2, ν = 1.5, α = 1.5 and β = 3.

Figure 9 :

 9 Figure 9: L q versus ν for p = 0, λ = 0.1, η = 1.5, µ = 10, α = 0.01, β = 0.5.

Figure 10 :

 10 Figure 10: L q versus ν for p = 0.5, λ = 0.1, η = 1.5, µ = 10, α = 0.01, β = 0.5.

Figure 11 :

 11 Figure 11: L q versus ν for p = 1, λ = 0.1, η = 1.5, µ = 10, α = 0.01, β = 0.5.

Figure 12 :

 12 Figure 12: E(L) versus ρ for p = 0, η = 3, ν = 1.

Figure 13 :

 13 Figure 13: E(L) versus ρ for p = 0.5, η = 3, ν = 1.

Figure 14 :

 14 Figure 14: E(L) versus ρ for p = 1, η = 3, ν = 1.

Acknowledgements

The authors would like to thank the referees for valuable comments and constructive suggestions that help to improve the presentation of the paper.

with

which can be written as

We note

so, we can write Eq.( 24) as follows:

The solution of this equation is as follows

where

Substituting F(z) in Eq.( 26), we get

To determine the constant k, we put z = 1 in Eq.( 27) and we use the equation below corresponding to the probability to have a busy server

then,

(vii) The mean busy period E(L) of the system is

Proof 4

The mean number of customers L q (respectively L s ) in the orbit (respectively in the system) given in (i)

(respectively in (ii)) under the steady-state condition are obtained by differentiating (20)

( respectively ( 21)

with respect to z and evaluating at z = 1.

The mean waiting time W q (respectively W s ) in the orbit (respectively in the system) give in (iii) (respectively in (iv)) is related to the mean number of customers L q (respectively L s ) in the orbit (respectively in the system) by the Little formula [START_REF] Little | A proof for the queuing formula: L = λ[END_REF]), L q = λW q and L s = λW s .

The probability R of having an empty system, given in (v), is obtained by summing the two probabilities P 0 and Q 0 given by ( 2) and ( 22).

(vi)The steady-state interrupted frequency N of the service caused by collisions given in Eq. ( 40), represents the sum of the interrupted frequency caused when the arriving customer from outside the system (primary customer) enters directly to the busy server and the interrupted frequency due to retrial of the customer from the orbit when the server is busy. Then

which can be rewritten as

By replacing Q(1) and Q ′ (1), given respectively by ( 14) and (3), we obtain N given in Eq. ( 38).

(vii) The busy period of the system, L, starts with the arrival of a primary customer who finds the system empty and ends at the first departure epoch in which the system becomes empty again.

From the regenerative processes theory, it is easy to get the following formula:

Using ( 22), we obtain E(L) given in Eq.( 39).


Lamia Lakaour
email: lakaour.lamia@gmail.com

Djamil Aissani

Karima Adel-Aissanou

Kamel Barkaoui
email: kamel.barkaoui@cnam.fr

Djamil Aïssani

D A Ïssani

retrial queue with collisions and transmission errors

Keywords: Retrial queue, Collisions, Transmission errors, Classical retrial rate, Generating function

Introduction

Queues with repeated attempts are characterized by the feature that arriving customers who find the server busy join a virtual group of blocked customers, called orbit, to try again for their requests in a random order and at random time intervals. Queues in which customers are allowed to conduct retrials have been widely used to model many practical problems in telephone switching systems, telecommunication networks and computers competing.

For the retrial queues, it is necessary to define the mechanism of retrials. The most usual mechanism described in the classical theory of retrial queues is the so-called classical retrial policy in which each source (packet, call, etc.) in orbit seeks service independently of each other after an exponentially distributed time with mean 1 η . Thus, in this policy the intervals between successive repeated attempts are exponentially distributed with rate jη, when the orbit size is j [START_REF] Aissani | Optimal control of an M/G/1 retrial queue with vacations[END_REF]. Another type of retrial policy, well known for the modeling of ALOHA protocol in communication networks, in which the time between two successive repeated attempts is controlled by an electronic device and consequently is independent of the number of customers applying for service. This type of retrial is named constant retrial policy, whose rate is (1δ j,0 )v, where δ j,0 denotes Kronecker's delta and j the number of repeated customers [START_REF] Fayolle | A simple telephone exchange with delayed feedbacks[END_REF]. In [START_REF] Artalejo | Steady-state solution of a single-server queue with linear repeated requests[END_REF], Artalejo and Gómez-Corral treat both models in a unified way by defining a linear retrial policy with rate (1 -δ j,0 )v + jη. The detailed overviews for retrial queue can be found in the bibliographies of Artalejo [START_REF] Artalejo | Accessible bibliography on retrial queues[END_REF]- [START_REF] Artalejo | Accessible bibliography on retrial queues: Progress in 2000-2009[END_REF], the surveys paper of Yang and Templeton [START_REF] Yang | A survey on retrial queues[END_REF], Falin [START_REF] Falin | A survey of retrial queues[END_REF], Choi and Chang [START_REF] Choi | Single server retrial queues with priority calls[END_REF] and Kim and Kim [START_REF] Kim | A survey of retrial queueing systems[END_REF], the books of Falin and Templeton [START_REF] Falin | Retrial queues[END_REF] and Artalejo and Gómez-Corral [START_REF] Artalejo | Retrial Queueing Systems: A Computational Approach[END_REF].

Retrial queues are widely discussed by researchers for modeling communication problems. However there are few who are interested in the problem of collisions. This phenomenon very widespread in telecommunications and whose management resolves many problems linked to the quality of service (QoS). Concretely, a collision occurs when two (or more) stations try to emit at the same time on the same channel. In the terminology of retrial queue, this phenomenon is translated by the fact that a customer arrives and finds the server busy, then the two customers (the arriving customer and the customer in service) join the orbit with a certain probability.

Queues with repeated attempts is found specifically in communication protocols modeling, such as CSM A/CD. Choi et al [START_REF] Choi | Retrial queues with collision arising from unslotted CSM A/CD protocol[END_REF] have discussed a retrial queuing model with collision and constant retrial rate arising from unslotted CSM A/CD (Carrier Sense Multiple Access with Collision Detection) protocol. They have derived the generating function of the limiting distribution of the number of customers in the retrial group when the channel is available. In [START_REF] Choi | Retrial queues with collision arising from unslotted CSM A/CD protocol[END_REF], the service time of the customer consists of two consecutive phases and the collision occurs when the arriving customer finds the channel busy with the first phase of a customer's service time. Kim [START_REF] Kim | Retrial queueing system with collision and impatience[END_REF] has considered the M/M/1 retrial queue with collision and impatience. Kumar et al [START_REF] Kumar | A single server feedback retrial queue with collisions[END_REF] have analyzed the feedback retrial queuing system with collisions and linear retrial policy. Kvach and Nazarov [START_REF] Kvach | Sejourn Time Analysis of Finite Source Markov Retrial Queuing System with Collision[END_REF] have considered the M/M/1//N retrial queue with collision. In [START_REF] Jailaxmi | Performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collision[END_REF], a performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collision has been considered. For those systems [START_REF] Jailaxmi | Performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collision[END_REF][13][15] [START_REF] Kvach | Sejourn Time Analysis of Finite Source Markov Retrial Queuing System with Collision[END_REF], if an arriving customer finds the server busy, then the collision occurs between the arriving customer and the customer in service resulting in both being shifted to the retrial group. In [START_REF] Kumar | A single server feedback retrial queue with collisions[END_REF], the authors have considered that the probability that primary customers accede to the service when it is either free or busy is the same. However, this is not realistic. This shows the necessity of taking two different probabilities in our work and more details will be given in the rest of the paper.

Several results have been reported on retrial queues with collision by considering a specified retrial policy depending on each particular application. However, the study of retrial queueing systems taking into consideration both the collision and transmissions errors is interesting, and we do not find much work in this respect in the literature. Based on this observation, we have investigated retrial queueing system with collisions, transmissions errors and classical retrial rate. In this paper, we consider the case where probabilities of joining the server by primary customer when the server is idle and busy respectively are different. It is equal to p when the server is free and (1 -θ) when the server is busy.

In this paper:

• At the customer's arrival from outside the system, if the channel is free, this customer accedes to the server with probability p or joins the orbit with probability (1 -p). In practice, for example, in the distributed coordination function (DCF) mode of IEEE 802.11, even if the channel is idle, the station doesn't transmit immediately, but transmits only if the server is idle for a period of time equal to DIFS (Distributed InterFrame Space). So, we have considered that the customer arriving from outside the system and finding an idle server accedes to the server with probability p;

• If the server is busy on the arrival of the primary customer, this customer collides with the customer in service and both will join the orbit with probability (1 -θ), or joins alone the orbit with probability θ;

• In addition to collisions, transmission errors are also taken into account. We have considered that the customer in service gets served without transmission error and leaves the system with probability γ or joins the retrial group for another attempt with probability (1 -γ).

In this study, we consider a retrial queue with collisions and transmission errors. The collision may occur when a primary arriving customer finds the server busy. In this case, both the primary customer and the customer in service will join the orbit, while transmission errors usually occur due to an erroneous packet or due to a non-ideal channel condition. This retrial queueing system can be applied for the modeling of the distributed coordination function (DCF) mode of IEEE 802.11 basic access scheme, where the loss and the re-transmission of packets can be due to either collisions or transmission errors. In IEEE 802.11, a station can not detect collisions or transmission errors while transmitting a packet, so the station can not interrupt its transmission. In order to notify the sender that the packet has been received successfully, a positive acknowledgment ACK is sent after receiving the packet.

The remainder of the paper is organized as follows. Section 2 presents the mathematical model description. Section. 3 provides the steady-state probabilities of the orbit size and the server status, and the stability condition of the system. In Section 4, we obtain interesting performances using steady-state analysis, while in Section 5, we give numerical examples to illustrate the theoretical results. Section 6 gives concluding remarks .

Model description

We consider an M/M/1 retrial queue with collisions and transmission errors. Primary customers arrive from outside the system according to a Poisson process with rate λ. Upon the arrival of the primary customer, if the server is free, this customer accedes to the server with probability p or joins the orbit with probability (1 -p). The inter-retrial time of each customer in the orbit is exponentially distributed with parameter η, i.e. we assume that the repeated attempts follow the classical retrial policy. The customer that enters in service (the primary customer/the customer from the orbit) gets served successfully (without transmission error) and leaves the system with probability γ (0 < γ ≤ 1) or joins the retrial group due to transmission error with probability (1 -γ). If the server is busy, the arriving customer (primary customer) collides with the customer in service resulting in both being shifted to the orbit with probability (1 -θ) or joins immediately alone the orbit with probability θ.

We assume that the service times of customers are independent, follow an exponential distribution with mean 1/µ and the stochastic processes involved in the system are independent.

The state of the system at time t can be represented by the pair (C(t), X(t)), where X(t) denotes the number of customers in the orbit and C(t) equals 1 if the server is busy and 0 if the server is free . It is clear that the process {C(t), X(t); t ≥ 0} is a continuous time Markov chain, irreducible, aperiodic and time-homogeneous with state space S = {(0, j), (1, j)/j = 0, 1, ...}.

Analysis of the steady-state distribution

Let P j = lim t→∞ P {C(t) = 0, X(t) = j} and Q j = lim t→∞ P {C(t) = 1, X(t) = j}, j ≥ 0. They represent the joint distributions of the server state and the orbit length (i.e., the number of customers in the orbit) at steady state.

1-p) λ 1-p) 1-p) 1-p) 1-p) 1-p) 0,0 0,1 0,2 0,j 1,0 1,1 1,2 1,j 2 3 j (j+1 ) ( j+2) ( 1 - ) µ ( 1 - ) µ ( 1 - ) µ ( 1 - ) µ ( 1 - ) µ ( 1 - ) µ
(0,j +1)

(1,j +1) Using the balance equations of the Markovian system illustrated in Fig. 1, we obtain

λ λ λ λ λ θλ p λ p p p p ϒμ ϒμ ϒμ ϒμ ϒμ ϒμ θ λ θ λ θ λ θ λ θ λ θ λ (1-θ) λ (1-θ) λ (1-θ) λ (1-θ) λ (1-θ) λ (1-θ) λ λ λ λ λ
λP 0 = γµQ 0 , (1)
(λ + η)P 1 = λ(1 -p)P 0 + (1 -γ)µQ 0 + γµQ 1 , (2)
(λ + jη)P j = (1 -p)λP j-1 + (1 -θ)λQ j-2 + (1 -γ)µQ j-1 + γµQ j , j = 2, 3, ..., (3)
(λ + µ)Q 0 = λpP 0 + ηP 1 , (4)
(λ + µ)Q j = λpP j + (j + 1)ηP j+1 + λθQ j-1 , j = 1, 2, 3, .... ( 5
)
The condition for the system stability is

λ < γµ. ( 6
)
We use the partial generating functions method defined below to solve Eqs. ( 1)-( 5)

P (z) = ∞ ∑ j=0 P j z j and Q(z) = ∞ ∑ j=0 Q j z j , z ∈ [0, 1], (7)
where P (z) (respectively Q(z)) is the P GF (Probability Generating Function) of the number of customers in the orbit when the channel is idle (respectively busy).

By multiplying both sides of Eqs. ( 1)-( 5) by z j and summing, we obtain these following two first order linear differential equations

ηzP ′ (z) + (λ -(1 -p)λz)P (z) = γµQ(z) + (1 -γ)µzQ(z) + (1 -θ)λz 2 Q(z), ( 8
) ηP ′ (z) + λpP (z) = λQ(z) + uQ(z) -λθzQ(z). ( 9
)
After multiplying both sides of Eq. ( 9) by z and subtracting from Eq. ( 8), we obtain

λP (z) = (γµ -λz)Q(z). ( 10
)
For z = 1, Eq. ( 10) becomes λP (1) = (γµ -λ)Q [START_REF] Aissani | Optimal control of an M/G/1 retrial queue with vacations[END_REF].

Under the stability condition (6), then P (1) + Q(1) = 1, it follows that the probability that the server is busy is

Q(1) = λ γµ , ( 11
)
the probability that the server is idle is

P (1) = 1 - λ γµ . ( 12
)
By differentiating Eq. ( 10) with respect to z and substituting into Eq. ( 8), we obtain the following differential equation

Q ′ (z) + [(θλ 2 -pλ 2 )z 2 + (pλγµ -λη -µλ -λ 2 )z] -ληz 2 + ηγµz Q(z) = 0. ( 13
)
Resolving [START_REF] Kim | Retrial queueing system with collision and impatience[END_REF] for Q(z) and after some computations, for z ∈ (0, 1], we get

Q(z) = λ γµ (γµ -λ) η+µ+λ-θγµ η e (p-θ)λ η (1-z) (γµ -λz) -η+µ+λ-θγµ η . ( 14
)
As z → 0+, Q(0) = Q 0 . Thus, from ( 14) and ( 1), we obtain

P 0 = (γµ -λ) η+µ+λ-θγµ η e (p-θ)λ η (γµ) -η+µ+λ-θγµ η . ( 15
)
By substituting ( 14) into [START_REF] Falin | Retrial queues[END_REF], we obtain the probability generating function P (z) of the orbit size when the server is free

P (z) = γµ -λz γµ (γµ -λ) η+µ+λ-θγµ η e (p-θ)λ η (1-z) (γµ -λz) -η+µ+λ-θγµ η . ( 16
)
The probability generating function for the number of customers in the orbit denoted K(z), is defined as P (z)+Q(z), then

K(z) = γµ -λz + λ γµ (γµ -λ) η+µ+λ-θγµ η e (p-θ)λ η (1-z) (γµ -λz) -η+µ+λ-θγµ η . ( 17
)
The probability generating function for the number of customers in the system H(z), defined as H(z) = P (z) + zQ(z) is given in Eq. ( 18)

H(z) = (γµ -λ) η+µ+λ-θγµ η e (p-θ)λ η (1-z) (γµ -λz) -η+µ+λ-θγµ η . ( 18
)

System characteristics

The mean number of customers L q (respectively L s ) in the orbit (respectively in the system) under steady-state condition are obtained by differentiating (17) (respectively ( 18)) with respect to z and then evaluating at z = 1

L q = - λ γµ - (p -θ)λ η + λη + λµ + λ 2 -λθγµ η(γµ -λ) , ( 19
)
L s = - (p -θ)λ η + λη + λµ + λ 2 -λθγµ η(γµ -λ) . ( 20
)
The mean number of customers L s and L q in the system and the orbit respectively at the steady state are related by the following formula

L s = L q + Q(1),
where Q( 1) is given in Eq. [START_REF] Fayolle | A simple telephone exchange with delayed feedbacks[END_REF].

The probability R of the orbit being empty is obtained by summing the two probabilities P 0 and Q 0 given by ( 1) and ( 15),

R = λ + γµ γµ (ηγµ -λη) η+µ+λ-θγµ η e (p-θ)λ η (ηγµ) -η+µ+λ-θγµ η . ( 21
)
The mean waiting time W q (respectively W s ) in the orbit (respectively in the system) is related to the mean number of customers L q (respectively L s ) in the orbit (respectively in the system) by the Little formula [START_REF] Little | A proof for the queuing formula: L = λ[END_REF], L q = λW q and L s = λW s

W q = - 1 γµ - p -θ η + η + µ + λ -θγµ η(γµ -λ) , ( 22
)
W s = - (p -θ) η + η + µ + λ -θγµ η(γµ -λ) . ( 23
)
The mean number of repeated attempts d by a customer given in Eq. ( 24) can be determined as follows

d = ηW q , d = - η γµ -p + θ + η + µ + λ -θγµ γµ -λ . ( 24
)
Another interesting system performance measure in this model of retrial queue is the steady-state interrupted frequency N of the service caused by collisions. This performance measure, given in Eq. ( 25), represents the collisions frequency caused when the arriving customer from outside the system (primary customer) accedes directly to the busy server, N = (1 -θ)λQ [START_REF] Aissani | Optimal control of an M/G/1 retrial queue with vacations[END_REF].

(25)

By replacing Q(1), by [START_REF] Fayolle | A simple telephone exchange with delayed feedbacks[END_REF], we obtain

N = (1 -θ)λ 2 γµ . ( 26
)
The busy period of the system L, starts with the arrival of a primary customer who finds the system empty and ends at the first departure epoch in which the system becomes empty again. From the theory of regenerative processes, it is easy to get the following formula for the expectation

E(L) = P -1 0 -1 λ , using (15)
, E(L) is obtained as

E(L) = 1 λ ((ηγµ -λη) -η+µ+λ-θγµ η e - (p-θ)λ η (ηγµ) η+µ+λ-θγµ η -1). ( 27
)

Numerical illustrations

In this section, we present some numerical examples to study the effect of the input parameters on the system characteristics.

In Figs. 234, to show the impact of θ ( (1 -θ) is the collision probability

)
, we plot P 0 , against η and that for three values of θ (θ = 0.2, θ = 0.6 and θ = 1). For this, we choose λ = 0.5, µ = 1 and γ = 1 and we take different values of p (p = 0, p = 0.5 and p = 1). We observe that P 0 increases as the value of the retrial rate η increases for any value of θ, but we observe that P 0 increases further as the probability θ increases. We report the same observation about the evolution of P 0 when we increase the value of p. These results are close to the real behavior of the system. In Fig. 5, we plot P 0 , against retrial rate η for three values of γ (γ = 0.5, γ = 0.7 and γ = 0.9); whereas the other parameters are fixed. When the probability γ becomes small, the number of customers (packets) that need retransmission due to transmission errors becomes more important, which generates a large mass of customers in orbit, which leads to a low probability of having an empty orbit. For p = 1, λ = 0.5 and µ = 1, Figs. 678, show how the number of customers L q in the orbit changes when the value of η increases. For this, we represent this impact for different values of γ (γ = 0.6, γ = 0.8 and γ = 1). In each figure, we fixe θ (θ = 0, θ = 0.5 and θ = 1). L q is a decreasing function of the the parameters η, γ and θ. In Fig. 9, we want to illustrate how the steady-state interrupted frequency N of the service caused by collisions vary when the value of θ increases. We do this illustration, for two values of λ (λ = 0.5 and λ = 0.9). In Fig. 10, we give an overview of how the mean busy period E(L) of the system changes when the value of η increases. This for different values of the probability θ. It can be seen from Fig. 9 that the steady-state interrupted frequency N of the service caused by collisions decreases considerably with respect to the probability θ. However, we observe that N = 0 when θ = 1. This is evident and consistent with our results. It is observed also in Fig. 10 that E(L) decreases as η and θ increase.

Conclusion

In this work, a Markovian single server retrial queue with collisions and transmission errors has been investigated. Using the Markov process theory, we have derived the steady-state equations. We have obtained the joint steady-state probability generating functions of the server state and the orbit length, and we have computed some important characteristics. Finally, to validate our model, we did some numerical illustrations. The numerical results obtained confirm our analytical investigations. This retrial queue can be used for the modeling and performance analysis of the Distributed Coordination Function (DCF) of IEEE 802.11 basic broadcast mode (basic access scheme). Our model is different from those studied in the literature in the fact that the probability to join the orbit when the server is free is different when the server is busy. The collision in this study is limited to primary customers that finds the server busy. In future work, we plan to study both collisions due to retrial of customers from the orbit and from outside the system.

Fig. 1

 1 Fig. 1 Transition diagram of an M/M/1 retrial queue with collisions, transmission errors and classical retrial policy.

Fig. 2 P

 2 Fig. 2 P 0 as function of η for p = 0. Fig. 3 P 0 as function of η for p = 0.5.

Fig. 4 P

 4 Fig. 4 P 0 as function of η for p = 1.

Fig. 5 P

 5 Fig. 5 P 0 as function of η for λ = 0.05 p = 0, θ = 1, µ = 1.

Fig. 6

 6 Fig. 6 Lq versus η for θ = 0. Fig. 7 Lq versus η for θ = 0.5.

Fig. 8

 8 Fig. 8 Lq versus η for θ = 1.

Fig. 9 N

 9 Fig. 9 N versus θ for η = 0, 01 µ = 1, p = 1, γ = 1.

Fig. 10 E

 10 Fig. 10 E(L) versus µ for λ = 0, 01 µ = 0.1, p = 1, γ = 1.

Acknowledgements

The authors would like to thank the referee for valuable suggestions and comments that help to improve the presentation of the paper.