Oualid Jouini
email: oualid.jouini@ecp.fr

• Auke Pot
email: pot@ccmath.com

Ger Koole
email: koole@few.vu.nl

Yves Dallery
email: yves.dallery@ecp.fr

Online Scheduling Policies for Multiclass Call Centers with Impatient Customers

Keywords: call centers, queueing systems, abandonments, online scheduling policies

We consider a call center with two classes of impatient customers: premium and regular classes. Modeling our call center as a multiclass GI/GI/s + M queue, we focus on developing scheduling policies that satisfy a target ratio constraint on the abandonment probabilities of premium customers to regular ones. The problem is inspired by a real call center application in which we want to reach some predefined preference between customer classes for any workload condition. The motivation for this constraint comes from the difficulty of predicting in a quite satisfying way the workload. In such a case, the traditional routing problem formulation with differentiated service levels for different customer classes would be useless. For this new problem formulation, we propose two families of online scheduling policies: queue joining and call selection policies. The principle of our policies is that we adjust their routing rules by dynamically changing their parameters. We then evaluate the performance of these policies through a numerical study. The policies are characterized by simplicity and ease of implementation.

Introduction

A call center, or in general a contact center, is defined as a service system in which customer representatives (agents or servers) serve customers (callers), over telephone, fax, email, etc. Managing a call center is a diverse challenge due to many complex factors related to uncertain and time-varying demand for service. In this paper we focus on a real-time problem, namely customer routing and agent scheduling.

The routing problem is a control problem which has received a lot of attention as a call center application. We consider here a V-model according to the canonical designs presented in [START_REF] Garnett | An Introduction to Skills-Based Routing and its Operational Complexities[END_REF]. We assume that all agents are flexible enough to answer all requirements of service. However, we divide customers into two different classes according to their importance, premium and regular customers. In addition, we assume that customers can abandon. It is natural in practice that a waiting customer is willing to wait for only a limited time, and will hang up within that time. Introducing abandonments in theoretical models is valuable, as ignoring abandonments leads to overstaffing and pessimistic estimations of queueing delays, see [START_REF] Duder | Towards "Zero Abandonments" in Call Center Performance[END_REF]. [START_REF] Garnett | Designing a Call Center with Impatient Customers[END_REF] show that models with and without abandonment tend to perform differently, even if the abandonment rate is low. Models including abandonments are therefore more realistic and will yield more accurate managerial insights.

Given a staffing level, our purpose is to develop control schemes for arrival calls and idle agents, subject to satisfying a constraint related to the probabilities of being lost (probability to abandon).

We notice that the abandonment probability is one of the major indicators used in practice. Another widely used indicator is the 80/20 rule where we stipulate that at least 80% of the customers should wait less than 20 seconds. Our objective is to meet a target ratio constraint between the achieved abandonment probabilities of the two customer classes. This problem formulation was inspired by a real call center problem. The reason behind it is to translate a desired fairness between customer classes. In our call center case, the abandonment of any class of customers is equivalent to a loss of goodwill. Both classes are indeed valuable for the company with a particular preference to the premium class. Having nearly no abandonments of the premium class and a lot of abandonments of the regular one is not desirable. The call center would instead prefer to have more abandonments of premium calls and fewer abandonments of regular ones. This is captured through a ratio of the abandonment probabilities. The ratio would be typically between 0 and 1. A low value of this ratio would translate to a strict preference of the company to premium calls. A ratio close to 1 would however translate an equal preference between the two classes. A value in between would translate a certain degree of preference.

In practice, managers traditionally handle this problem by separately setting for each customer class a constraint on the probability to abandon. The drawback of such a formulation is that it is too much dependent on some predicted workload. Once the actual workload deviates from the predicted one, we are no longer able to meet the predefined performance constraints, and we could well reach an undesired preference between customer classes. Indeed we often use static strict priority rules, such as a static strict non-preemptive priority for one class over the other. So if the workload is underestimated, most of the capacity of the system will be dedicated to premium calls.

We may then satisfy the performance constraint of premium calls, while having a heavily penalized one for regular calls. However if the workload is overestimated, the performance of premium calls will be very high and that of regular ones will not profit that much from the overcapacity.

Several studies, notably [START_REF] Jongbloed | Managing Uncertainty in Call Centers using Poisson Mixtures[END_REF] and [START_REF] Avramidis | Modeling Daily Arrivals to a Telephone Call Center[END_REF], have shown that the arrival process and the workload are hard to predict in call centers. A new formulation of the routing problem using a target ratio constraint between the service levels of the two classes would, as a consequence, be a better alternative. It allows to better control the different situations which may occur (under-or overestimation of the workload). Satisfying the constraint ratio enables to share as desired the capacity of the system between the two classes. In addition, this new formulation generalizes the traditional one where we have a target abandonment probability for each class.

We use online policies in order to reach our objective. An online policy is a discipline that is continuously updated as customers arrive and are processed. The main advantage of our control policies is that they require no information about the arrival processes in advance, i.e., the method is data-driven. If fluctuations in workload occur, the capacity allocated between the two classes is changed. The parameters of the policies are automatically adopted in such a way that the target on the service levels is met.

We focus on developing simple and useful online routing policies that are based on priority schemes. The provision of differentiated service levels often relies on the use of different priorities between customers. [START_REF] Schrage | The Queue M/G/1 with the Shortest Remaining Processing Time Discipline[END_REF] showed that scheduling policies similar to those in multiclass priority queues allow to achieve high performance measures, often nearly as good as those under optimal policies. We derive various schemes for online assignment of customers to queues, as well as for selection to service of waiting customers. The policies we propose are characterized to be workconserving (non-idling), which is natural for large service systems such as call centers. A policy is defined to be non-idling if there can be no idling servers when there are waiting customers.

In our opinion, the restriction does not decrease the usefulness of the analysis because, in practice, Automatic Call Distributors (ACDs) do not often support idling policies, such as thresholds or reservations policies. Under a threshold policy, customers are selected for service by following some predefined rules that are based on the system state. Under a reservation policy, some servers are only allocated to a given type of customers. This kind of policies would force one or more servers to be idle while there are waiting customers. This paper has three major contributions. The first contribution is the introduction of a new class of call routing policies which are robust to changes in the workload. The second is the analysis of the effect of scheduling policies on various performance measures which contributes to the literature on multiclass queueing systems. The third contribution is to propose simple and efficient scheduling policies that satisfy the target ratio constraint. This is especially important because it is known that in practice a non-intuitive solution with a complex structure would never be used. The analysis yields quantitative insights, as well as useful principles and guidelines for the control problem.

The rest of the paper is organized as follows. In Section 2 we review the literature related to our work. In Section 3 we present the problem under consideration. Sections 3.1 and 3.2 are devoted to formulate the call center model and to define the notations, respectively. Section 3.3 concretely formulates the control problem. Section 3.4 gives some structural results that will help us later in the understanding of the behavior of the scheduling policies. In Section 4 we develop two families of online scheduling policies that allow to satisfy the target ratio constraint. The family of queue joining policies is addressed in Section 4.1, and the family of call selection policies is addressed in Section 4.2. In Section 5 we present and discuss simulation experiments of the proposed policies.

The paper ends with some concluding remarks and highlight some directions for future research.

Literature Review

There is an extensive and growing literature on call centers, and in general on contact centers. We refer the reader to [START_REF] Akşin | The Modern Call-Center: A Multi-Disciplinary Perspective on Operations Management Research[END_REF] for a survey of the recent literature on call center operations management. The literature related to this paper spans mainly two areas. The first deals with queueing systems with impatient customers. The second deals with the control of queueing systems in general, and the control of the V-model in particular.

In the following, we highlight some of the literature with regard to the first area. Queueing models incorporating impatient customers have received a lot of attention in the literature. To underline the importance of the abandonment modeling in the call center field, the authors in Mandelbaum and Zeltyn (2008) give some numerical examples that point out the effect of abandonment on performance measures. [START_REF] Brown | Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective[END_REF] conduct an empirical study to characterize the distribution of abandonment times. The literature on queueing models with abandonments focuses especially on performance evaluation. We refer the reader to [START_REF] Ancker | Queueing with Impatient Customers Who Leave at Random[END_REF], [START_REF] Garnett | Designing a Call Center with Impatient Customers[END_REF], and references therein for simple models assuming exponential abandonment times. [START_REF] Koole | A Formula for Tail Probabilities of Cox Distributions[END_REF] develops an algorithm for calculating tail probabilities of Cox distributions.

This can be useful to compute many performance measures in a queueing system with abandonments. In [START_REF] Garnett | Designing a Call Center with Impatient Customers[END_REF], the authors study the subject of Markovian abandonments. They suggest an asymptotic analysis of their model under the heavy-traffic regime. Their main result is a relation between the number of agents, the offered load and system performance measures, such as the probability of delay and the probability to abandon. This can be seen as an extension of the results of [START_REF] Halfin | Heavy-Traffic Limits for Queues with Many Exponential Servers[END_REF] by adding abandonments. [START_REF] Whitt | Efficiency-Driven Heavy-Traffic Approximations for Many-Server Queues with Abandonments[END_REF] establishes the Efficiency-Driven many-server heavy-traffic limits for a Markovian queue with abandonments and limited waiting line. Other papers assumed abandonments to follow a general distribution. [START_REF] Whitt | Fluid Models for Multiserver Queues with Abandonments[END_REF] uses fluid models to approximate the steady state performance measures of queueing models with generally distributed interarrival, service and abandonment times. Other studies include those by [START_REF] Baccelli | On Queues With Impatient Customers[END_REF], [START_REF] Brandt | Assymptotic Results and a Markovian Approximation for the M(n)/M(n)/C + GI System[END_REF], [START_REF] Ward | A Diffusion Approximation for a Markovian Queue with Reneging[END_REF], [START_REF] Whitt | Engineering Solution of a Basic Call-Center Model[END_REF], [START_REF] Avramidis | Optimizing Daily Agent Scheduling in a Multiskill Call Center[END_REF], and references therein.

Let us now focus on the second area of literature close to our work, i.e., the control of queueing systems. Scheduling policies have been studied in great depth within the context of queueing systems. A scheduling policy, or a discipline of service, prescribes the order in which customers are served. [START_REF] Randolph | Queueing Methods for Services and Manufacturing[END_REF] classifies scheduling policies into those using online schedule rules and those using static schedule rules. Each of the above classes of policies can be further classified into two major classes: agent scheduling and customer routing, see [START_REF] Garnett | An Introduction to Skills-Based Routing and its Operational Complexities[END_REF] for more details. In the following, we present results about scheduling policies under the framework of V-models. [START_REF] Pekoz | Optimal Policies for Multi-Server Non-Preemptive Priority queues[END_REF] addresses the analysis of a multiserver non-preemptive priority queue with exponentially distributed interarrival and service times. She finds and evaluates the performance of an asymptotically optimal policy that minimizes the expected queueing delay for high priority customers. [START_REF] Guérin | Queueing-Blocking System with two Arrival Streams and Guard Channels[END_REF] presents a model without waiting queues. The model contains a multiserver station, which receives low and high priority arrivals. He develops an admission policy for the low priority customers such that the fraction of blocked high priority customers is bounded and he analyzes the system under that policy. In the context of call centers, Gurvich et al. (2008) consider a large-scale system under the V-design and characterize asymptotically optimal scheduling and staffing schemes (as system load grows to infinity). The optimal scheduling and staffing schemes minimize the staffing costs subject to satisfying quality of service constraints for the different customer classes. [START_REF] Maglaras | Pricing and Design of Differentiated Services: Approximate Analysis and Structural Insights[END_REF] consider profit maximization for a loss system twoclass V-model with pricing, sizing, and admission control. [START_REF] Milner | Service-Level Agreements in Call Centers: Perils and Prescriptions[END_REF] explore the role that service level constraints in outsourcing contracts play in settings where the contractor firm has both contractual and non-contractual customers. Another paper that consider a similar idea of contract and noncontract customers is [START_REF] Bhulai | A Queueing Model for Call Blending in Call Centers[END_REF]. For a detailed survey of relevant papers considering the optimal control of the V-model, we refer the reader to [START_REF] Gurvich | Design and Control of the M/M/N Queue with Multi-Class Customers and Many Servers[END_REF]. We finally refer the reader to [START_REF] Gurvich | Service-Level Differentiation in Many-Server Service System via Queue-Ratio Routing[END_REF] and [START_REF] Dai | Optimal Control of Parallel Server Systems with Many Servers in Heavy Traffic[END_REF] for recent references on online policies developed within an economic context.

Framework

In this section, we first describe the basic model of our call center. Second, we define the various performance measures we are interested in. We then formulate the problem for which we want to propose the scheduling policies. Finally, we develop some structural results about the relation between the performance measures under consideration and various scheduling policies.

Model Description

We model our call center as a queueing system with two customer classes: a premium customer class A, and a regular one B. The model consists of two infinite queues, say queues 1 and 2, and a set of s parallel, identical servers representing the set of agents. All agents are able to answer all customer classes. The call center is operated in such a way that at any time, any customer can be addressed by any agent. Upon arrival, a customer is addressed by one of the available agents, if any. If not, the call joins one of the queues.

We consider two families of scheduling policies: queue joining and customer selection families.

We will describe in details the proposed policies among the two families in Section 4. In what follows we describe their functioning in general. A policy belonging to the first family of queue joining policies determines the rule of assigning customers upon arrival to one of the queues. Upon arrival, a customer of any class can be sent to any queue. That is, each time an arrival A or B enters the system, an individual decision is made as a function of the system state: assign this new arrival to queue 1 or 2. In other words, we want to specify here that there is no an a priori fixed rule of assigning for example all customers A to queue 1 and all customers B to queue 2. So queue 1 or 2 may contain a mix of the two customer types. Finally note for this first family that customers waiting in queue 1 have a non-preemptive strict priority over those in queue 2.

A policy belonging to the second family of customer selection policies determines at each service completion which waiting customer class should start service. For this family, there is no fixed priority between the two queues. Upon arrival, all customers A are sent to queue 1 and all customers B are sent to queue 2. At each service completion, the server who has just become idle will choose to serve one of the queues, or equivalently one of the customer classes. This decision is not static and is dynamically made as a function of the system state. In other words, if it happens at some times that we would like to improve the performance of class A, so an idle server would tend to choose to serve customers waiting in queue 1 (customers A). The opposite is done if it happens during other times that we would like to improve the performance of class B.

For both families of policies, customers waiting in a given queue, are served in the order of their arrivals, i.e., under FCFS. Also, the priority rule between the queues is non-preemptive because it is not common in call centers to interrupt the service of a customer and serve another one with a higher priority.

Interarrival times and service times are assumed to be i.i.d. and follow a general distribution.

In certain cases, we shall consider the exponential distribution for successive service times. The mean service time rates of customer classes A and B are µ A and µ B , respectively. In addition, we let the customers be impatient. After entering the queue, a customer waits a random length of time for service to begin. If service has not begun by this time she will abandon (leaves the queue). Patience times of classes A and B are assumed to be i.i.d. and exponentially distributed with rates γ A and γ B , respectively. Assuming identical distribution of patience within each class, independently from their position in the queue, seems to be a plausible assumption for call centers, see [START_REF] Gans | Telephone Call Centers: Tutorial, Review, and Research Prospects[END_REF]. Indeed, the tele-queueing experience in call centers is fundamentally different from that of a physical queue, in the sense that customers do not see others waiting and need not be aware of their "progress" (position in the queue) if the call center does not provide information about queueing delays. We impose the Markovian assumption on patience times in this paper. This is mainly to preserve tractability in our models. Though not all empirical studies from real call centers suggest that patience times are exponential (see for example [START_REF] Brown | Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective[END_REF]), considering this assumption allows us at least to capture the uncertainty in patience times and helps us as a consequence to gain useful insights.

The system is workconserving, i.e., an agent is never forced to be idle while customers are waiting. Finally, retrials are ignored, and abandonments are not allowed once a customer starts her service. We also do not allow jockeying between separate queues. Following similar arguments, the behavior of this call center can be viewed as a modification of a GI/GI/s + M queueing system.

The symbol M after the + indicates the Markovian assumption for times before abandonments.

Note that owing to abandonments, the system is unconditionally stable. One proof which also holds for a more general model is as follows. Consider an arbitrary GI/GI/s + GI queueing model (with general patience times) and let us prove that it is stable, i.e., the stochastic process representing the number of customers in system has a limiting distribution as time goes to infinity.

The number of customers in this GI/GI/s + GI model can be upper bounded by the number of busy servers in an associated GI/GI/∞ infinite server model. The service times in the associated GI/GI/∞ model have the same statistical distribution as that of the summation of service and patience times in the original GI/GI/s + GI model. The idea of the upper bound comes from the fact that the waiting time of a queued customer in the GI/GI/s + GI model is upper bounded by her patience threshold. So her sojourn time in system is upper bounded by her patience threshold plus her service time. The infinite-server model is known to be stable. We refer the reader to [START_REF] Whitt | On the Heavy-Traffic Limit Theorem for GI/G/infinity Queues[END_REF] for a discussion and further references. This finishes the proof of the stability of the GI/GI/s + GI model, and in particular our GI/GI/s + M model.

Notations

We denote by m the class of a customer, m ∈ {A, B}. We assume that at time zero the system starts empty. For a particular sample path, let n m (t) be the number of class m arrivals during the time interval [0, t], t > 0. Consider now a scheduling policy π and let a m π (t) be the number of class m customers that abandon the queue, and b m π (t) the number of those that are receiving service and those that have been already served in [0, t]. We consider for each class m, the fraction of customers that abandon. We also consider this performance for all customer classes (an arbitrary customer).

The fraction of class m customers that abandon during [0, t] is defined by

P m π (t) = a m π (t) n m (t)
. The fraction of abandonments during [0, t] for all classes is defined by

P π (t) = a A π (t)+a B π (t) n A π (t)+n B π (t) .
In the long run, the fraction of abandonments (probability to abandon) of class m customers, say P m π , and that of all classes, say P π , are given by P m π = lim t→∞ P m π (t) and P π = lim t→∞ P π (t). Recall that due to abandonments the system is stable so that the latter limits do exist (see the end of Section 3.1). We are now ready to define our main performance measure which we denote by c π .

It is the ratio of the probability to abandon of class A over that of class B, i.e.,

c π = P A π P B π .
(1)

Similarly we define the expected waiting time in queue of class m customers and that of all customer classes. Note that we only define these quantities for the customers who enter service.

Under a given scheduling policy π, let w m q,π (i, t) be the waiting time in queue of the i th class m customer who enters service, 0 ≤ i ≤ b m π (t). As in the usual way, the expected waiting time in the queue, say

W m q,π (t), during [0, t] of class m customers is defined by W m q,π (t) = b m π (t) ∑ i=1 w m q,π (i, t) b m π (t)
. and that of all customer classes, say W q,π (t), by

W q,π (t) = b A π (t) ∑ i=1 w A q,π (i, t) + b B π (t) ∑ i=1 w B q,π (i, t) b A π (t) + b B π (t)
. In the long run, the expected waiting time in the queue of served class m customers, say W m q,π , and the overall expected waiting time in queue of all served customers, say W q,π , are given by W m q,π = lim t→∞ W m q,π (t), and W q,π = lim t→∞ W q,π (t).

We also define the squared difference of waiting times of served class m customers and that of all classes, in the transient regime as well as in the long run. The squared difference of the waiting time

of served customers of class m, say V m π (t), during [0, t] is given by V m π (t) = b m π (t) ∑ i=1 ( w m q,π (i, t) -W m q,π (t)
) 2 b m π (t) As for the squared difference of the waiting time in queue of all served customers within [0

, t], it is defined by V π (t) = ∑ m∈{A,B} b m π (t) ∑ i=1 ( w m q,π (i, t) -W q,π (t) ) 2 ∑ m∈{A,B} b m π (t)
In the long run, the squared difference of class m served customers, say V m π , and the overall squared difference of all served customers, say V π , are given by V m π = lim t→∞ V m π (t), and V π = lim t→∞ V π (t). In the numerical experiments shown later, we will consider the standard deviation defined as the square root of V m π and denoted by σ m π , and that of V π denoted by σ π .

Problem Formulation

In this section we motivate and formulate our objective with regard to the scheduling policies we aim to develop. Due to the highly uncertain environment of call centers, it is usually hard to estimate the workload within a relative accuracy. We often end up in practice with either an underestimated, or an overestimated workload. Next, the common practice in call centers is to develop routing policies that aim to reach different service level constraints for different customer classes. In this paper we focus on service levels in terms of the abandonment probabilities. Having these statements in hand, one may obviously raise the following issue. In case of a fixed number of agents, the abandonment probability of each customer class will be affected by the forecasting error and will thereafter deviate from the predefined one. Furthermore, the actual service level deviations can be very different between the customer classes (for example when using strict priority scheduling policies). This behavior is undesirable for a call center manager because one would lose some given fairness between customer classes. For any work condition, a manager would like to reach a desired fairness between customer classes.

Based on this motivation, we formulate the following problem. We assume that scheduling has already taken place, such that the number of available agents is known. We aim to develop scheduling policies that satisfy a target ratio constraint, say c * , of the abandonment probabilities between of the two customer classes. In mathematical terms, we look, if at all possible, for π ∈ Π subject to

c π = c * , ( 2
)
where Π denotes the class of workconserving non-preemptive scheduling policies.

The target ratio c * translates a desired preference between the two customer classes that we want to reach for any actual workload. As we will see later by means of Theorem 2, the c * target formulation generalizes the traditional formulation where we have a service level constraint for each class. If the workload is quite correctly estimated, having a target ratio of abandonment probabilities is equivalent to having a target abandonment probability for each class. In addition if the workload is incorrectly estimated (under-or overestimated), the capacity of the system is shared over customer classes in a way that allows to preserve (if possible) the predefined preference between customer classes. Specifically, we are interested in policies which are easy to implement in practice and which require as little information as possible about the arrival process and the capacity of service.

The expected waiting times of customer classes are not included in the problem formulation.

The drawback of minimizing the waiting times of the served customers is that it could result in policies that are very similar to LCFS (per customer class), which is considered to be unfair.

Further, we do not consider neither the abandonment probability, nor the expected waiting time in queue of an arbitrary customer. When considering workconserving policies, these quantities are unchanged for identical distributions of service times. They are furthermore not much sensitive when the distributions do differ. We will discuss these statements in the next sections in detail.

Non-workconserving policies are ignored because they would make the routing problem even more complicated. Moreover, this kind of policies is not often seen in practice.

Structural Results

In this section we investigate the impact of scheduling policies on various performance measures.

These results would be helpful in the understanding of the behavior of the policies we will develop and in assessing their efficiency.

Consider first queueing systems with infinitely patient customers, that is, a customer never leaves the queue. It is well known (see for example [START_REF] Gross | Fundamentals of Queueing Theory[END_REF]) that the expected remaining workload in the queue is independent of the queueing discipline, assuming that the remaining total service or work required is order-of-service independent during any busy period. In other words, no service needs are created or destroyed within the system, implying: no abandonment in the middle of a service, no preemption when service times are not exponentially distributed, no forced idleness of servers, and so on. The proof can easily be done by comparing the diagrams of the cumulative work for two different queueing disciplines during the busy period, elsewhere both systems behave identically due to the workconserving property.

By means of Theorem 1, we motivate why workconserving policies are considered in this paper.

Theorem 1 concerns a GI/GI/s + M system, which has i.i.d. and generally distributed interarrival and service times, s servers, and exponentially distributed patience times. The proofs of all results from this section are given in Section 1 of the supplementary material Jouini et al. (2010).

Theorem 1 Consider a non-preemptive GI/GI/s + M queue and consider policies that do not depend on service time realizations. Then there is a workconserving policy that minimizes the abandonment probability, P.

The memoryless condition of patience times in Theorem 1 is a necessary and sufficient condition when preemption is not allowed. Suppose that the abandonment occurs always during the beginning of the customer waiting, such under the decreasing failure rate distributions for example, then it can be optimal to wait for a new arrival and serve it before it abandons instead of serving a customer who has waited already for some time. In what follows, we derive some results related to workconserving policies. In Theorems 2 and 3, we investigate the conservation of the abandonment probability and the expected waiting time in the queue with respect to the scheduling policies, respectively. Some consequences are derived in Corollaries 1 and 2.

Theorem 2 Consider a GI/GI/s + M queue. Patience times are assumed to be i.i.d. and exponentially distributed. Then the abandonment probability P is unchanged for all workconserving non-preemptive scheduling policies that are not allowed to depend on the service time realizations.

Note that the result in Theorem 2 does not hold if service times depend on the order of service, or if we allow preemption when service times are not exponentially distributed, or if patience times are not identically and exponentially distributed. In the case of exponential service times, proving that P is unchanged for all workconserving policies with or without preemption can be obtained using Lemma 2 in [START_REF] Jouini | Monotonicity Properties for Multiserver Queues with Reneging and Finite Waiting Lines[END_REF]. In the latter, the authors have proved this result for a GI/M/s/K + M queue with a limited waiting space.

In Theorem 3, we focus on the waiting time in the queue with respect to workconserving nonpreemptive scheduling policies. We again consider a GI/GI/s + M queue, and focus on three different definitions of the expected waiting time. Let W q be the expected waiting time in queue of served customers. Let W ab q be that of abandoning ones, i.e., the expected sojourn time in the queue before leaving the system without being served. Finally, we define W tot q as the expected overall waiting time in the queue for all customers, i.e., served as well as abandoning customers. In Theorem 3, we prove an intuitive result for the conservation of W tot q . In addition, we give the scheduling policies that minimize and maximize W q and W ab q . Although the abandonment probability as shown in Theorem 2 does not vary for any workconserving non-preemptive scheduling policy, W q and W ab q do vary.

Theorem 3 Consider a GI/GI/s + M queue. Patience times are assumed to be i.i.d. and exponentially distributed. When considering the class of workconserving non-preemptive scheduling policies, the following holds 1. W tot q does not depend on the scheduling policy.

2. W q and W ab q depend on the scheduling policy.

3. The upper (lower) bound of W q is achieved under the FCFS (LCFS) discipline of service.

4. The upper (lower) bound of W ab q is achieved under the LCFS (FCFS) discipline of service.

Note that although the first moment W tot q does not depend on the discipline of service, the second moment of the overall waiting time, and thus the full distribution, does depend on the discipline of service. As shown in Theorem 3, the maximum of the expected waiting time for served customers, W q , is achieved under FCFS discipline. However we conjecture based on a well-known property from the literature (see for example [START_REF] Gross | Fundamentals of Queueing Theory[END_REF]) that the minimum of its variance is also achieved under the FCFS policy. In practice, if the value of W q under the FCFS policy is almost equal to the value of other policies, a call center manager usually prefers the FCFS policy because of its fairness. We refer the reader to Avi-Itzhak and Levy ( 2004) for more details on the fairness property in queueing systems.

The result in Theorem 3 is still valid when considering also preemptive scheduling policies.

However, service times have to be exponentially distributed. We finally comment that the second statement in Theorem 3 is still valid for generally distributed service times.

Corollary 1 Consider a GI/GI/s + M queue with two customers classes A and B. Service times as well as patience times are identically distributed for both customer classes. Then the overall abandonment probability P and the overall expected waiting time W tot q are constant for any workconserving non-preemptive scheduling policy.

In the case of a two-class queue, we denote by π A and π B the policies that give strict nonpreemptive priority to class A and class B customers, respectively.

Corollary 2 Consider a GI/GI/s + M queue with two of customers classes A and B. Service and patience times are identically distributed for both customer classes. Then, for any workconserving non-preemptive policy π the ratio of the abandonment probabilities c π satisfies the following relation

c π A ≤ c π ≤ c π B , ( 3
)
where c π A and c π B are the ratios in the long run under π A and π B , respectively.

Note that values of c π outside the interval [c π A , c π B ] may be achieved through non-workconserving policies, such as thresholds or reservations policies. These policies are indeed useful to discriminate between customer classes. Under such policies, the lower bound for the abandonment probability P A (P B ) is equal to the value achieved under the policy that gives strict preemptive priority to class A (B) customers. Obviously, the upper bound for P A or P B is 1. It is reached for a given class by simply refusing to serve customers of that class.

With the waiting time being defined as the waiting time of all customers (including both served and abandoned customers), we next give Theorem 4 which only holds for Markovian patience times. Note also that this theorem holds even for GI/GI/s + M queues that are working under non-workconserving policies.

Theorem 4 Consider a GI/GI/s + M queue, with two customer classes A and B, and working under a given scheduling policy π. Service and patience times are identically distributed for both customer classes. Let c π be the long run ratio of the abandonment probabilities. Then the long run ratio of the expected waiting times is also c π .

Online Scheduling Policies

The call center we consider allows for flexible scheduling through dynamically sequencing the order of service, hereafter referred to as online scheduling. With the current technology, this is also doable for most call centers. However, an interesting and challenging problem is to design scheduling policies that are structurally simple and easy to implement.

In this section we develop online scheduling policies that allow, if at all possible, to the target ratio constraint c * . We consider techniques that do not anticipate the future and that are based on the history of the system, i.e., the achieved ratio c at the moment of the decision. These can be classified as online updating methods and online routing. Without loss of generality, we only consider objective ratios satisfying c * < 1. The case c * = 1 boils down to the FCFS discipline of service. The reason is that our model working under FCFS (within each class and for both classes)

is simply equivalent to a single class queue working under FCFS. The case c * > 1 is not relevant for our analysis because class A denotes the premium calls. Furthermore, even if we would like to investigate that case, it suffices to apply the analysis for the case c * < 1 by exchanging class A by class B and vice versa.

We propose online scheduling policies that belong to two different families. The first is the family of queue joining policies and is presented in Section 4.1. The second is the family of call selection policies and is presented in Section 4.2.

Queue Joining Policies

We propose three queue joining scheduling policies, denoted by π 1 , π 2 and π 3 . Upon a customer arrival, a policy determines a rule for the queue assignment. We assume that customers in queue 1 have a non-preemptive priority over customers in queue 2. Customers of a class can be sent to any queue. We recall that our policies do not anticipate on future events. They just react to the realization of the ratio that is determined by the history of the process. A comparison analysis of these policies is thereafter addressed in Section 5, using simulation experiments.

Scheduling Policy π 1 : The scheduling policy π 1 starts identically to a strict priority policy that gives the higher priority to class A customers. Scheduling Policy π 3 : The scheduling policy π 3 starts identically to policies π 1 and π 2 until the first customer B finishes service. Following again the same notations as before, let a new arrival enter the system. Under π 3 , a customer B is always routed to queue 2. However, the assignment rule of customers of class A is as follows. If c k ≥ c * , then a new class A arrival is routed to queue 1, otherwise if c k < c * , it is routed to queue 2. An illustration of π 3 is shown on Figure 3.

The scheduling policy π 1 can be immediately obtained intuitively. It allows the achieved ratio to be updated upon each arrival such that it converges in the long run to the objective. The idea behind π 2 is that we keep always customers of class A in the high priority queue, however when it is necessary, we assign customers of class B to this queue to improve their service level (which deteriorates the service level of customers of class A). Such a rule allows to increase the transient ratio and keep it close to the objective. As a consequence, the ratio converges in the long run to the desired value. Policy π 3 can be viewed as another variant. It sometimes allows to penalize class A customers by assigning them to the low priority queue, which again allows to increase the In Conjecture 1, we consider an M/GI/s + M queue (Markovian interarrival times) and we conjecture some properties related to the achieved ratios.

Conjecture 1 Using the above notations, the following holds.

π 1 reaches c * if and only if c π

A ≤ c * ≤ c π B . 2. π 2 reaches c * if and only if c π A ≤ c * ≤ 1.

π 3 reaches c * if and only if c π

A ≤ c * ≤ 1.
In what follows, we give an intuitive explanation of why we expect this conjecture to hold.

Let us consider the first statement of Conjecture 1 and let us take our basic model working under the scheduling policy π 1 . Because we are considering workconserving non-preemptive policies, the lower and upper bounds of the achievable ratios are c π A and c π B , respectively. It is clear that if we apply policy π A (strict priority to class A) starting at time zero, we will reach the ratio c π A in the long run. We will also reach c π A in the long run if we apply π A starting at any time t > 0.

The reason is related to the fact that the number of customers in system, at the epochs of service completions, is a renewal process (Poisson process of arrivals and i.i.d. service times). The above remark is also valid for policy π B (strict priority to class B) and the upper bound c π B . These are the two extremes. Consider now c * ranging between c π A and c π B . At time t > 0, if it happens that the achieved ratio is strictly lower than c * , then giving priority to customers of class B (which is possible under π 1 ) allows necessarily to go beyond c * after a given duration of time. Continuing in doing this, the ratio will converge to c π B . In the other case, if it happens at time t that the achieved ratio is strictly greater than c * , then giving priority to customers of class A (which is possible under π 1 ) allows necessarily to go below c * after a given duration of time. Continuing in doing this, the ratio converges to c π A . So, π 1 makes the achieved ratio fluctuating below and above c * . Continuing applying these manipulations, we conjecture that the ratio of the abandonment probabilities will converge to c * . This has been confirmed by simulation experiments.

Consider now the second statement. Let c F CF S be the achieved ratio under the FCFS policy.

With the same explanation as above, we can see that any ratio ranging from c π A to c F CF S can be reached by π 2 . On the one hand, The lower bound for P A is achieved when we give strict priority to class A. In addition at the same time, the upper bound for P B is reached. Thus, the minimum possible achievable target ratio corresponds to policy c π A . On the other hand, the lower bound for P B is achieved by assigning all class B arrivals to queue 1, which also allows to achieve the upper bound for P A . This corresponds to the FCFS policy for arrivals of all classes. Hence, the achieved ratio could not be worse than that under the FCFS policy, c F CF S = 1. Finally, we note that the explanation of the third statement is similar to that of the second one. This finishes the discussion about Conjecture 1.

One may construct several auxiliary policies similar to the ones above. For example, instead of changing the priority rule at each new arrival epoch, we only change it at the arrival epoch of the customer who finds all servers busy and both queues empty. Then, we continue that rule until the end of the current busy period. With regard to reaching the target ratio, the latter class of policies has the same properties as those of π 1 , π 2 and π 3 . One drawback could be that they are less reactive to correct the transient ratio. A further possibility is to construct similar policies by changing the priority rule cyclically; at a given arrival and based on the transient ratio, we decide for the priority rule and we apply it for a given fixed number of arrivals. Once the cycle finishes, we determine the priority rule at the epoch of the arrival that follows the cycle. Again, we keep that rule for the same given fixed number of new arrivals, and so on.

Call Selection Policies

In the following we focus on a dual family of policies, namely the family of call selection policies.

We assign all class A (class B) customers to queue 1 (queue 2). On the contrary to the previous class of policies, we do not implement a fixed priority discipline between the queues. Customers waiting in the queues are selected by using so-called waiting time factors, as described in [START_REF] Lu | Scheduling to Minimize General Functions of the Mean and Variance of Sojourn Times in Queueing Systems[END_REF]. These selections occur after each service completion of an agent, when both queues are non-empty. The choice about the next customer to serve depends on the waiting time factors, of which exactly one is associated with each queue. When selecting a customer for service, the first customer in each queue is considered. From these two customers, we serve the customer for which the product of its waiting time and the waiting time factor is the highest. The waiting time factors are only queue-dependent. This family of policies has appealing properties: it takes the actual waiting times of the customers into account, is fair with respect to the order of service within each class, and the reactivity of the policy is expected to be high. Note that setting both waiting time factors equal results in a policy that serves customers from the two classes within in a FCFS order, and by taking one factor equal to 0, one of the classes has full priority over the other, independent of the waiting times.

Let us denote the waiting time factors by α A and α B for queues 1 (waiting class A customers) and 2 (waiting class B customers), respectively. The waiting time factors are dynamically adjusted by a method having parameter β, with β ∈ [0, 1]. At each epoch of a service completion, say t, one of the waiting time factors is set to β, while the other one is set to 1. Thus, there are two possibilities at t, namely (α A (t), α B (t)) ∈ {(1, β), (β, 1)}. At time t, the abandonment probabilities of the two classes, P A (t) and P B (t), are computed by considering the historical events. Then, we can formulate the proposed policy as

(α A (t), α B (t)) ← { (1, β) if P A (t) ≥ c * P B (t) (β, 1) if P A (t) < c * P B (t) . ( 4
)
We do not claim that this is the best way to update the waiting time factors. This would require a more thorough analysis which out of the scope of this paper. An immediate advantage of the current method is a high response in fluctuating transient scenarios. More comments about this policy are given in Section 5. In the same way as that in the explanation of Conjecture 1, we conjecture here that waiting time factors as in Equation ( 4 in the right way. In the limit case when β approaches 1, swapping will not allow to force the priority for one given class. So, the objective c * can no longer be reached (c(t) will converge to 1 which corresponds to FCFS).

Alternatively, another method for updating the waiting time factors is to keep α A fixed over time and to update α B dynamically. Under the assumption that α A is fixed, it is straightforward to decrease α B when P A is high, and increase α B otherwise. A difficulty is to develop a method that updates α B in a simple but effective way, without many control parameters. We investigated several possibilities by means of numerical experiments. As a result, we decided to decrease the parameter α B by multiplication by η and increase it by division by η, with 0 < η ≤ 1. The factor α B was updated after each event of service completion.

Simulation Experiments

The nature of the scheduling policies we are considering in this paper (online policies) makes a tractable analysis too complicated. Both analytical and numerical methods are hard to be derived. We thereafter resort to simulation experiments in order to prove the efficiency of the proposed policies and gain useful guidelines. We consider numerical examples of Markovian systems: exponential interarrival, service and patience times. Note that in practice, service times of premium calls are likely to be the largest because the call center would pay more attention to them by giving more personalized answers, µ A < µ B . For patience times, premium customers should be in most call center cases more patient than regular customers because the formers know that they will experience a good quality of answer after their waiting, γ A < γ B .

We consider 6 systems, denoted by System 1, . . ., System 6. Systems parameters are chosen such that we get realistic scenarios. In all systems the number of servers is s = 50. The service rates are µ A = 0.15 and µ B = 0.25. The abandonment rates are γ A = 0.3 and γ B = 0.35. From one system to another, we vary the total arrival rate so as we get different "service utilizations", 5, 6, 7, 9, 11, and 13, respectively. The "service utilization" is increasing starting from 100% in System 1 until 260% in System 6. On purpose we choose highly loaded systems in order to see the effect of abandonment. We consider balanced cases for the arrival processes in order to avoid including further aspects that may complicate the understanding of the simulation results. Recall that abandonments make our systems unconditionally stable. The simulations are done for the target ratios c * = 0.5, 0.7 and 0.9. We determined for each system the interval [c π A , c π B ], and we checked that the values c * = 0.5, 0.7 and 0.9 are ranging in all of these intervals. We notice for the balanced examples (λ A = λ B ) considered here we have

λ A +λ B sµ . We choose, λ A = λ B =
c π A = 1 cπ B .
For queue joining policies, we present the performance measures under policies π 1 , π 2 and π 3 , as well as those under policy π A (high priority for class A customers). For call selection policies, we give the performance measures for β equal to 0, 1/4, 1/2, 3/4, and 1.

Experiments for the Queue Joining Policies

The simulation results for c * = 0.7 are presented in Tables 123456. In Tables 13-18 and 19-24 in Section 2 of the supplementary material Jouini et al. (2010), the results are presented for c * = 0.5 and 0.9, respectively. The rows, corresponding to the quantity c, are to indicate the achieved long run ratio under each scheduling policy.

= λ B = 9, c * = 0.7
We see from the experiments that the target ratio is always met by policies π 1 , π 2 and π 3 , which agrees with Conjecture 1. For each system, the value of the ratio under policy π A represents a lower bound for the achievable ratio under any workconserving non-preemptive scheduling policy.

We can not do better when considering that class of policies.

= λ B = 13, c * = 0.7
We further analyze in this section the expected and the standard deviation of the waiting time in queue of each class of served customers. A rigorous proof of our claims is out of the scope of this paper. We only give some general ideas and intuitive explanations to support the claims we derive.

We especially focus on the standard deviation of the waiting time. It has often been argued that a system with reasonable and predictable waiting times may be more desirable than a system with a lower expected waiting time but a higher variance, see [START_REF] Lu | Scheduling to Minimize General Functions of the Mean and Variance of Sojourn Times in Queueing Systems[END_REF] for more details.

For class A customers, starting from the lower value, most experiments show that the standard deviation values are ordered according to policies π A , π 2 , π 3 and π 1 . The reason is basically related to the well-known property in queueing theory which claims that the FCFS discipline minimizes waiting time variance (time in queue and in system) when the queueing discipline is service time independent. We refer the reader to [START_REF] Randolph | Queueing Methods for Services and Manufacturing[END_REF] for a more extensive discussion. The best we can do for customers of class A under a workconserving non-preemptive policy is not to give at any time the priority to customers of class B. Such a situation allows the realizations of class A waiting times to be minimized. This is the case for policy π A . Next, since the discipline of service within queue 1 is FCFS, π A should lead to the lower variance. With regard to the order of service of customers of class A, policy π 1 deviates more than π 2 and π 3 from the FCFS discipline. This tells us that π 1 has the highest variance. When comparing policies π 2 and π 3 , one may see that on the contrary of policy π 3 , policy π 2 respects the FCFS order for customers of class A, which indicates a lower variance than under policy π 3 . The experiments for class A show also that the expected waiting times in queue are ordered according to policies π A , π 1 , π 3 and π 2 . The order π 1 then π 3 then π 2 is expected because of the general property that FCFS maximizes the expected waiting time of served customers. Policy π A is the best for the expected waiting time of served customers. An explanation would be related to the small values of waiting times achieved under that policy.

For class B customers, starting from the lower value, we conclude from the majority of the experiments that the standard deviation values are structured for policies π 3 , π 2 , π 1 and π A .

When comparing policies π 1 , π 2 and π 3 , the explanation is identical to that conducted for the first comment. As for policy π A , the only explanation we have is related to the waiting time values. The larger waiting times of class B customers are achieved under policy π A because of their lower priority. This might explain why that policy has the highest variance for waiting times of class B customers, because generally higher values have also higher variance. In addition to that, uncertainty in interarrival, abandonment and service times, allows to have a non-zero probability that some customers of class B enter service without waiting or within only a short delay. This makes their waiting times variance to be the highest. From the experiments, we also see for class B that the expected waiting times in queue of served customers are ordered according to policies π 1 , π 2 , π 3 and π A . One may explain these results through the same arguments used above.

We notice that we only gave some directions to compare the policies with regard to the expected value and standard deviation of waiting times. For instance, the comparison should lie in the values of λ A and λ B , also in the target ratio constraint c * . A target ratio close to 1 would make our policies work in a similar manner than that of the FCFS discipline, whereas an objective far from 1 (being in the two queues is close to a FCFS policy. Such high values of β do not allow to sufficiently discriminate one class of customers to the detriment of the other. Both classes have almost equal priority of service. In addition, the parameter β c * (introduced in Section 4.2) is increasing in c * .

The explanation is as follows. As c * increases, giving the priority to one class over the other one is less and less needed. This allows β c * to be higher. In the limit case (for c * = 1), β c * reaches its upper bound (β c * = 1).

We see also from the experiments that the overall expected waiting time of the served customers increases as β increases. The reason pertains to the fact that as β increases, the scheduling policy approaches the FCFS policy. As shown in Theorem 3, the FCFS policy maximizes the expected waiting time in queue of served customers. Meanwhile, the expected waiting time of the abandoned customers decreases. Finally, the variance of the waiting times (for all classes) is decreasing in β.

One intuitive explanation is due to the known property of the FCFS policy, which minimizes the waiting times variance. As β increases, the behavior of our policy approaches that of the FCFS policy, and as a consequence, the overall variance decreases. In the limit case (β = 0), we have an alternation for service selection which is the most distant from the FCFS policy. Not surprisingly, its corresponding variances are the highest.

It is also interesting to observe that β c * is independent of the workload. One intuitive explanation would be related to the balanced cases we consider here, λ A = λ B = λ. In such cases, increasing λ will increase the number of arrivals of both classes by the same factor. For a given objective ratio c * , choosing one β in the same interval [0, β c * ] allows thereafter to increase the number of abandonments of both classes by the same factor, which allows to reach c * . The reason is that we keep unchanged the way we are giving priorities to customer classes. If we choose β beyond β c * , we have no longer the sufficient flexibility to discriminate customer classes so as we increase the abandonments of both classes in the same way as the arrivals.

Comparison

In what follows we address the comparison of the two families of policies: queue joining and call selection policies. We use the simulations results to draw the basic conclusions.

Not surprisingly the expected waiting times of the served customers are shorter for the queue joining policies. This conclusion is valid for each class of customers and for all classes. The reason has to do with the order in which customers are scheduled for service. Under the call selection policies, customers of each class are served in the order of their arrival. This makes the waiting times larger under that type of policies (see Theorem 3). As a consequence, the expected waiting time of all served customers is also the highest under the call selection policies. The second conclusion is that the variances (equivalently the standard deviations) of the waiting times are lower for the call selection policies. The explanation is identical to the previous one and is again pertaining to the order of service of customers.

One may also compare both types of policies from the reactivity perspective. We mean by reactivity, the speed of a given policy to make the transient ratio c(t) close to the objective.

Although both types of policies allow to satisfy the target ratio constraint in the long run, we comment that they have different reactivities. The call selection policies are more reactive. One intuitive explanation is as follows. When for example the transient ratio is lower than the objective, both policies give high priority to class B in order to increase this transient ratio. A queue joining policy as π 2 will assign a new customer B to the queue with higher priority, however it may happen that there are type A waiting customers in that queue ahead of the customer B of interest. In such a case, the transient ratio would thereafter still decrease. It would increase once we finish to serve the type A customers waiting in the highest priority queue. This case does not occur under the call selection policies. Depending on the circumstances, the latter policies immediately allow to increase or decrease the transient ratio, by scheduling one given class for service to the detriment of the other class.

The reactivity of the call selection policy strongly depends on the value of β. It is maximized for β = 0 such that customers of the class that allows to push the transient ratio to c * is served first. For higher values of β the waiting times of the customers are more taken into account such that customers that have waited longer than others would have a higher priority for service. This can yield actions that are sometimes not optimal with respect to the ratio constraint. By means of the waiting time factors, a trade-off is made between serving the customer with the longest waiting time and serving the customer that improves the transient ratio of abandonment probabilities.

Sometimes the optimal actions of both objectives will be the same but not in all cases.

Conclusions

We focused on a fundamental real-time problem of call centers management. We considered a two-class call center and developed online scheduling policies subject to satisfying a target ratio constraint of the abandonment probabilities of the two customer classes. This new formulation of the control problem is robust with respect to the system workload. Furthermore, it generalizes the traditional formulation where we have a target abandonment probability for each class. The policies are argued to be relevant in practice; they are efficient, easy to understand for managers, predictable and easy to implement.

First, we gave some structural results in order to better understand the impact of different scheduling policies on the performance measures of interest. Second, we proposed several online scheduling policies allowing to meet the considered constraint. Third, we conducted a simulation study in order to compare the proposed policies with regard to several performance measures such as the expected and the variance of the waiting time in queue of served customers.

In this paper, we focused the analysis on a given period of the day. In future research we would like to focus on intervals of a day. Then it would be interesting to find a method that translates the whole day objective into a set of objectives per period of the day. It would be also interesting to extend the analysis to more than two customer classes and agents with different skill sets.

  class A, it is routed to queue 1, and if it is class B, it is routed to queue 2. An illustration of π 1 is shown on Figure1.Scheduling Policyπ 2 : The scheduling policy π 2 starts identically to π 1 until the epoch at which the first customer B finishes service. Following the same notations as in the last paragraph, let a new arrival enter the system. Under π 2 , a customer A is always routed to queue 1. However, the assignment rule of class B customers is as follows. If c k < c * , a new class B arrival is routed to queue 1, otherwise if c k ≥ c * , it is routed to queue 2. An illustration of π 2 is shown on Figure 2.

  Figure 1: Scheduling policy π 1

  ) reach any objective between the two limiting cases where one of the two classes has full priority, namely c π A and c π B . In addition, we conjecture that for each feasible objective c * , there exists β c * ∈ [0, 1] so that choosing any value in [0, β c * ] and assigning it to the parameter β allows to reach c * . Any value beyond β c * could not allow to reach c * . Just think of what happens near the boundary β = 0. In this case, swapping the factors α A (t) and α B (t) will give each time the full priority to one of the classes such that the transient ratio fluctuates around c * . When β increases, swapping the waiting time factors does no longer necessarily guarantee that the right class gets the priority so that the transient ratio moves

  After the epoch at which the first class B customer finishes her service, we apply the following assignment rule for any new arrival (denoted by the k th arrival). Let d k be the epoch of that arrival. Let P A k (P B k ) be the achieved service level until d k for class A (B). Let c k be the achieved ratio starting until d k , c k = P A k /P B k . If c k < c * , then we give high priority to class B, i.e., if the new arrival is class A, it is routed to queue 2, otherwise, it is routed to queue 1. However if c k ≥ c * , we give high priority to class A, i.e., if the new arrival is

Table 1 :

 1 λ A = λ B = 5, c * = 0.7

		π A	π 1	π 2	π 3		π A	π 1	π 2	π 3
	c	0.203	0.700	0.700	0.700	c	0.160	0.700	0.700	0.700
	P A	3.915%	8.462%	8.483%	8.434%	P A	7.479%	19.233% 19.160% 19.191%
	P B	19.249% 12.089% 12.119% 12.048%	P B	46.803% 27.476% 27.372% 27.415%
	P	11.583% 10.276% 10.302% 10.240%	P	27.137% 23.355% 23.267% 23.302%
	W A	0.128	0.254	0.273	0.270	W A	0.249	0.585	0.667	0.658
	W B	0.487	0.297	0.314	0.324	W B	1.440	0.690	0.759	0.819
	W	0.292	0.275	0.293	0.296	W	0.684	0.635	0.711	0.734
	σ A	0.187	0.484	0.373	0.383	σ A	0.253	0.827	0.522	0.565
	σ B	0.754	0.567	0.483	0.467	σ B	1.286	0.980	0.705	0.681
	σ	0.557	0.527	0.431	0.427	σ	0.986	0.904	0.617	0.628

Table 2 :

 2 λ A = λ B = 6, c * = 0.7

		π A	π 1	π 2	π 3
	c	0.155	0.700	0.700	0.700
	P A	10.851% 28.507% 28.502% 28.523%
	P B	70.096% 40.725% 40.717% 40.748%
	P	40.488% 34.613% 34.609% 34.635%
	W A	0.365	0.864	1.062	1.040
	W B	2.724	0.968	1.190	1.340
	W	0.958	0.911	1.120	1.176
	σ A	0.327	1.142	0.602	0.696
	σ B	1.750	1.396	0.799	0.815
	σ	1.377	1.264	0.701	0.767

Table 3 :

 3 λ A = λ B = 7, c * = 0.7

		π A	π 1	π 2	π 3
	c	0.219	0.700	0.700	0.700
	P A	20.473% 41.288% 41.273% 41.262%
	P B	93.598% 58.982% 58.961% 58.946%
	P	57.039% 50.134% 50.113% 50.104%
	W A	0.723	1.227	1.691	1.633
	W B	5.500	0.924	1.926	2.139
	W	1.079	1.102	1.788	1.841
	σ A	0.509	1.536	0.740	0.889
	σ B	2.628	1.803	0.840	0.957
	σ	1.526	1.658	0.791	0.951

Table 4 :

 4 λ A

Table 5 :

 5 λ A = λ B = 11, c * = 0.7

		π A	π 1	π 2	π 3
	c	0.426	0.700	0.700	0.700
	P A	42.588% 55.095% 55.074% 55.084%
	P B	99.949% 78.702% 78.754% 78.695%
	P	71.284% 66.897% 66.912% 66.888%
	W A	1.784	1.974	2.542	2.534
	W B	8.563	0.519	3.134	3.153
	W	1.791	1.506	2.732	2.733
	σ A	0.659	1.365	0.918	0.932
	σ B	4.263	0.608	0.623	0.633
	σ	0.701	1.358	0.880	0.895

Table 6 :

 6 λ A

Acknowledgements

The authors would like to express their gratitude to Rabie Nait-Abdallah from Bouygues Telecom for several useful discussions which were at the origin of the problem analyzed in this paper.

under or beyond) would make the policies similar to the strict priority policy. Based on the analysis here, we can not distinguish a best policy. However, one may recommend policy π 2 . First, it reaches the objective ratio. Second, it gives (in most cases) the lowest variance of waiting times of customers A. Third, it allows to have a "good" variance for customers B, as well as for all customers.

Experiments for the Call Selection Policies

The simulation results are presented in Table 7-12 below for c * = 0.7. In Tables 25-30 and31-36 of the supplementary material Jouini et al. (2010), the results are presented for c * = 0.5 and 0.9, respectively. The rows (corresponding to the quantity c) display the achieved long run ratio under different values of β. The rest of this section is devoted to discuss the simulation results.

From the experiments, we obviously see that all quantities increase when the workload increases.

If β is high, for example 3/4 or 1, then the ratio c * is not reached. The reason is that the weights of the waiting times are almost equal, such that the discipline of service of the customers waiting


IMA Journal of Management Mathematics, special issue Operations Management in Service Systems

The service sector is the largest sector of the economy in most industrialised nations, and is fast becoming the largest sector in developing nations as well. Driven by today's new business environment including advanced telecommunications, accelerated business globalisation, increased automation, and highly on-demand and competitive innovations, the complexity of the operations management of service systems is continuously increasing. Managers of service systems are wrestling to deliver both traditional conflicting objectives of low operating costs and high service quality.

Consequently, the problems of operations management in service systems have recently attracted a lot of attention. The most popular applications relate to Services for Supply Chains, Health Care Systems and Call Centres. Given the size and the complexity associated with their operations, service systems have emerged as a fertile ground for academic research. We were delighted therefore to be offered the opportunity to edit this special issue of the IMA Journal of Management Mathematics on operations management in service systems. We hope that through this special issue we have achieved our objective which is to advance the current state of knowledge in this area, adding in parallel significant value to relevant real-world practices. This special issue demonstrates that all the key areas addressed in Manufacturing Operations (such as demand forecasting, acquiring capacity, deploying resources, and managing delivery) are now also reflected in the Services Operations agenda. However, issues such as behavioural factors, the emerging need for an increased standardisation and automation of the operations, handling technological advances (such as the digitalisation of systems) and cultural issues (such as those resulting from the globalisation of systems, e.g. big call centres) that introduce many interesting research opportunities have not yet sufficiently explored and more remains to be done in this area to gain new interesting managerial insights.

The call for this Special Issue has attracted many submissions. All manuscripts underwent a rigorous review process which resulted in the six papers presented in this special issue. This process required the cooperation of both authors and reviewers and we thank all of them for their efforts and diligence; they all made our role as Special Issue Editors easy and enjoyable.

The first two papers deal with applications to healthcare systems. Call centres operations are investigated in the third paper and the last three papers deal with supply chain applications. The papers included in this special issue span from 'hard' OR methodological approaches to fully qualitative studies. The first paper by Griffiths, Knight and Komenda focuses on bed management in a Critical Care Unit (CCU). The authors propose a mathematical model that shows how improvements in bed management may be achieved by distinguishing between two categories of patients: the unplanned (emergency) and the planned (elective) patients. The novelty of this paper lies in its moving away from the steady state conditions by assuming time dependent ones which is the case for the activities in all hospital environments. Banditori, Cappanera and Visintin develop in their paper a combined optimisation-simulation approach to the master surgical scheduling problem. The authors present a mixed integer programming model that produces a solution that indicates, for each day of the month and for each time slot of the day, the number of cases to treat and the surgery group these cases must belong to. The authors also describe a combined optimisationsimulation approach that allows fine-tuning of the optimisation model to trade off robustness and efficiency. The third paper by Saccani takes an integrated view on demand forecasting in call centres. An action research study in a multi-client call centre organisation is presented. The findings allow the suggestion of some design principles for the four dimensions involved in the forecasting process: method, organisation, people and technology. A forecasting framework is developed which is shown to account for substantial operational improvements. Bouzembrek, Allaoui, Goncalves and Baklouti develop in their paper a possibilistic linear programming (PLP) approach for supply chain network design with imprecise inputs: market demands, supplied quantities, transportation costs, opening costs, treatment and storage costs, which are modeled as fuzzy variables. A method for solving the PLP problem with fuzzy parameters is developed and its application to a European textile company is explored. The paper by Yu and Ramanathan investigates the links between business environment, employee competencies, operations strategy and performance for a sample of retail companies in China. The authors show that employee competencies fully mediate the effects of the business environment on operations strategy, and that the employee competenciesperformance connection is mediated by operations strategy. Finally, Altay is concerned with the efficiency and effectiveness of resources allocation in a 'disaster' situation. He develops an integer programming model for the capability-based multi-resource multilocation allocation problem. The model is intended to take advantage of a nationwide resource inventory listing in order to allocate response personnel, facilities, supplies and equipment to impact jurisdictions.

We hope you will find the papers included in this special issue interesting and thoughtprovoking and that they will stimulate your research and inform your practice.

Mohamed

  Zied BABAI BEM-Bordeaux Management School 680 cours de la Libération, 33405 Talence Cedex, France tel: +33 5 56 84 63 51 email: mohamed-zied.babai@bem.edu Oualid JOUINI Ecole Centrale Paris Grande Voie des Vignes, 92 290 Châtenay-Malabry, France tel: +33 1 41 13 15 02 email: oualid.jouini@ecp.fr


O Chavoya-Aceves
email: chavoyao@gmail.com

A possible generalization of quantum mechanics to special and general relativity

Keywords: interpretations of quantum mechanics, special relativity, quantum gravity

Based on a reinterpretation of Hamilton-Jacobi equation, a generalization of Madelung's hydrodynamic model of quantum mechanics is proposed, which is valid in the realm of special relativity and can be extended to study gravitational fields with quantum effects. We estimate that gravitational quantum effects will not be noticeable but for particles of very small mass at very high energy ≈ 1.223 × 10 19 G eV, for which the de Broglie wave-length is of the order of Planck's length. As an additional result, we show that there are elements to presume that, in fact, quantum mechanics is not a complete theory of motion.

Introduction

In classical analytical mechanics, the solution of the Hamilton-Jacobi equation appears as means to the end of finding a canonical transformation under which the corresponding Hamilton function is zero and, as a consequence, the new phase coordinates are constant. If H(q, p) and K(Q, P ) are the Hamilton functions of a mechanical system (for two systems of canonical coordinates) the equations of motion can be obtained from the conditions:

δ t f t0 p i dq i -Hdt = 0 or δ t f t0 -q i dp i -Hdt = 0. ( 1
)
and

δ t f t0 P i dQ i -Kdt = 0 or δ t f t0 -Q i dP i -Kdt = 0. ( 2
)
1

There are thus four possibilities in regards to the relation among the differnetial forms that appear in those integrals: p i dq i -Hdt = P i dQ i -Kdt + dF 1 p i dq i -Hdt = -Q i dP i -Kdt + dF 2 -q i dp i -Hdt = P i dQ i -Kdt + dF 3 -q i dp i -Hdt = -Q i dP i -Kdt + dF 4 [START_REF] Madelung | Quantentheorie in hydrodynamischer form[END_REF] where F 1 , F 2 , F 3 and F 4 are known as generating functions of the first, second, third, and fourth kind, respectively.

For a generating function S(q i , P i , t), of the second kind

p i = ∂S ∂q i ; Q i = ∂S ∂P i and K = H + ∂S ∂t , (4)
which suggests to look for a generating function satisfying the equation

∂S ∂t + H q i , ∂S ∂q i = 0; (5)
Then K ≡ 0, because of the third of equatins (4), and the phase coordinates (Q i , P i ) are constants of motion, or integrals of the canonical equations.

A complete solution of (5) has the form S = S(q 1 , ..., q n , t, P 1 , ..., P n ).

From it, an implicit solution of the canonical equations can be obtained, in terms of 2 • n + 1 constants (Q i , P i , t 0 ), as follows from [START_REF] Landau | Quantum Mechanics-non-relativistic theory[END_REF].

In this paper we look at the Hamilton-Jacobi equation from the perspective of a hydrodynamic analogy to classical mechanics. We obtain some results which we consider as evidence that quantum mechanics does not reduce to classical mechanics in the limit h ≈ 0 but to a particular case of it, which we take as an indication that it is not a complete theory of motion.

Those results allow us to propose a generalization of classical quantum mechanics, that though it abandons the concept of a wave function, it includes quantum mechanics as a particular case. This extension can, in turn, be generalized to special relativity, providing an example of a possible way to understand the interaction of matter and radiation in a self-consistent fashion. We express conservation of mass-energy in terms of a stress-energy tensor and the formalism makes not use of wave functions, but of the mathematical apparatus of tensor calculus. Therefore, the formalism can be translated to the language of general relativity. However we do not pretend to have a self-consistent general field theory of matter, radiation, and gravitation because our results refer to a single kind of material particles.

Hydrodynamic analogy of classical mechanics

Let's consider a classical Hamiltonian system with n degrees of freedom. The equations of motion in the phase space (q i , p i ) are

qi = ∂H ∂p i ; ṗi = - ∂H ∂q i (6)
Notice that in these equations the subindex can take any value in the set {1, • • • , n}. We prefer not to make that explicit in some of our equations to avoid unnecessary clutter. We focus our attention on a region R(0) of the configuration space q = (q 1 , ..., q n ) where a vector field p = p i (q 1 , • • • , q n ) is defined. In other words we are not referring to individual paths in the phase space, but to continuous families of paths, indexed by their coordinates at t = t 0 We assume this field to be continuous and differentiable as much as required to ensure the validity of our arguments.

As times passes and the phase points with the initial conditions (q, p(q)) at t = 0 move R = R(t) changes, as well as the field p = p(q, t) which is now a function of position in the configuration space and time. The Hamilton's function is thus a function of position in R(t) and time: H(q, p(q, t)).

The time derivative of p i along the phase trajectory that passes trough the point (q, p(t))-a convective derivative-is

Dp i Dt = ∂p i ∂t + j=n j=1 qj ∂p i ∂q j = ∂p i ∂t + j=n j=1 ∂H ∂p j ∂p i ∂q j .
Also, using [START_REF] Landau | The classical theory of fields[END_REF]:

Dp i Dt = - ∂H ∂q i + n j=1 ∂H ∂p j ∂p j ∂q i .
The reason to include the last term in the last equation is that, in [START_REF] Landau | The classical theory of fields[END_REF], all the phase variables are treated as independent whilst, in the case we are considering, the momenta are supposed to be functions of position in the configuration space and time. From the last two equations we get

∂p i ∂t + j=n j=1 ∂H ∂p j ∂p i ∂q j = - ∂H ∂q i + j=n j=1 ∂H ∂p j ∂p j ∂q i .
and by reordering the terms

∂p i ∂t + ∂H ∂q i = j=n j=1 ∂H ∂p j ∂p j ∂q i - ∂p i ∂q j . (7)
If it happens that there is a function φ(q) such that at t = 0

p i (q) = ∂φ ∂q i , (8)
the right side of (7) will be zero at t = 0; and this will not change as time passes, because of Helmholtz's Circulation Theorem [1, p. 180]. Therefore, if [START_REF] Chavoya | An explanation of spin based on classical mechanics and electrodynamics[END_REF] is true, [START_REF] Chavoya | An explanation of the quantum mechanics of particles[END_REF] can be simplified to

∂φ ∂t + H q i , ∂φ ∂q i = 0.
This equation is essentially [START_REF] Holland | The quantum theory of motion[END_REF], now presented as a particular case of [START_REF] Chavoya | An explanation of the quantum mechanics of particles[END_REF]

p j qj -H = L,
where L is the Lagrange function, which in this scenario is a function of position in the configuration state and time. This identifies φ as the mechanical action S, but for a function of time, which is of no consequence for the determination of the momenta using [START_REF] Chavoya | An explanation of spin based on classical mechanics and electrodynamics[END_REF]. Howver, there is an essential difference: Dφ Dt is a convective derivative.

Classical particle acted by a conservative force

If the system is a single particle acted by an external conservative field, the Hamilton function, in Cartesian coordinates, assumes the well known form

H(r, p) = p 2 2m + V (r); (9)
and ( 7) can be written as

∂p i ∂t + ∂ ∂x i p 2 2m + V = p j m ∂p j ∂x i - ∂p i ∂q j or ∂p ∂t + ∇ p 2 2m - p m × (∇ × p) = -∇V, (10)
and

m ∂v ∂t + (v • ∇)v = -∇V. (11)
(Notice the analogy to a perfect incompressible fluid, but for the condition ∇ • v = 0 that follows from the continuity equation.)

If ∇× p = 0 at t = 0 then. as aforementioned, there is a function S = S(r, t) such that p = ∇S and

∂S ∂t + (∇S) 2 2m + V (r) = 0. ( 12
)
The last equation is a particular case of (10) and it poses no difficulty to imagine initial conditions p = p(r, 0), where ∇ × p = 0. It follows then thatin principio-for the solution of any problem of the sort we are studying, (10) must be considered as the fundamental law and any theory-or description of motion-that is based on (12) must be understood as applicable to particular cases and, subsequently, insufficiently general, or incomplete.

As we have stressed in a previous paper [START_REF] Chavoya | Logical refutation of the Einstein-Podolsky-Rosen argument[END_REF] we consider a theory complete if:

1. The representation used by the theory is complete and the corresponding interpretation is unambiguous;

2. the axioms are logically consistent;

3. the theory does not leave any sensitive question without a sensitive answer;

4. the theory is true.

Madelung substitution in quantum mechanics

In 1927, one year after E. Schrödinger published his paper reducing the problem of quantization to an eigenvalue problem, E. Madelung published a paper revealing an analogy between classical hydrodynamics and quantum mechanics [START_REF] Madelung | Quantentheorie in hydrodynamischer form[END_REF].

Madelung made the substitution

Ψ = √ ρe iφ (13)
in Schrödinger equation for a single particle

ih ∂Ψ ∂t = - h2 2m ∆Ψ + V Ψ. ( 14
)
Because √ ρ can fail to be differentiable at points where ρ = 0, even if ρ is differentiable at those points, we will use Landau's substitution ([4, p. 52])

Ψ = Ae iφ , (15)
where A and φ are real functions.

Working on the terms in (14) one by one, and considering the real and imaginary parts

hA ∂φ ∂t + h2 2m A(∇φ) 2 + AV - h 2 2m ∆A = 0,
and

∂A ∂t + h m ∇A∇φ + h 2m A∆φ = 0.
By making S = hφ, these equations can be rewritten as

∂S ∂t + (∇φ) 2 2m + V - h2 2m ∆A A = 0. ( 16
)
and

∂ρ ∂t + 1 m ∇ • (ρ∇S) = 0. ( 17
)
Equation ( 16) corresponds to (12) with an extra term in the potential energy

Q = - h2 2m ∆A A (18)
known as the quantum potential. This, altogether with the equation p = ∇S is at the foundation of the de Broglie-Bohm pilot wave theory according to which [5, p. 62]:

...

1. An individual physical system comprises a wave propagation in space and time together with a point particle which moves continuously under the guidance of the wave.

2. The wave is mathematically described by ψ(x, t), a solution to Schrödinger's wave equation. ... Equation (16 is also used to prove that quantum mechanics corresponds to classical mechanics in the limit h = 0 [4, p. 52].

In our view, whether quantum mechanics should lead to (12) or the more general equation (10) in the classical limit, is a sensitive question wanting a sensitive answer, as required by the principle of correspondence. In regards to this we have two possibilities to consider: I If quantum mechanics should lead to (10) in the classical limit, the representation used by quantum mechanics, the wave function, is not complete, in the sense that it cannot be used to describe all physical possibilities. Correspondingly, quantum mechanics as it is currently formulated is not a complete theory of motion.

II If quantum mechanics should lead to (12) in the classical limit, the fact that this requires ∇ × p = 0 requires an explanation which is not found in Quantum mechanics and, accordingly, quantum mechanics is not a complete theory.

We do not see any a priori reason to prefer (12) instead of (10) as the classical limit of quantum mechanics. Equation ( 5) is a mathematical contrivance to find solutions to the canonical equations of motion, while (10) has physical meaning. In addition, see our argument in section 4, in particular equations (33) and (34). For these reasons, we feel compelled to consider what seems to be the simplest generalization of the equations of quantum mechanics that is compatible with (10) in the classical limit:

∂p ∂t + ∇ p 2 2m + V - p m × (∇ × p) = ∇ h2 2m ∆A A , (19)
∂A 2 ∂t + 1 m ∇ • A 2 p = 0, (20)
Those equations were obtained by E. Madelung [START_REF] Madelung | Quantentheorie in hydrodynamischer form[END_REF], who then abandoned them because he noticed that they do not generalize to the case of many particles. However, we still have [START_REF] Chavoya | An explanation of the quantum mechanics of particles[END_REF] of which ( 5) is only a particular case. Therefore, our question makes sense for a system of two or more particles.

Equation ( 19) is singular at those points where A = 0. This singularity is removed by multiplication of both sides of it with

ρ = A 2 because ρ ∂ ∂x i ∆A A = - ∂A ∂x i ∂ 2 A ∂x 2 j + A ∂ 3 A ∂x 2 j ∂x i = ∂ ∂x j A ∂ 2 A ∂x i ∂x j - ∂A ∂x i ∂A ∂x j .
Equation (19) takes then the form

ρ ∂p ∂t + ∇ p 2 2m + V - p m × (∇ × p) = -∇ • T , (21)
where

T ij = h2 2m ∂A ∂x i ∂A ∂x j -A ∂ 2 A ∂x i ∂x j .
Equation (21) simplifes to:

ρ ∂p ∂t + ρp m • ∇ p + ∇V = -∇ • T .
Using (20) we get:

∂ρp i ∂t + ∂Π ij ∂x j = 0, (22)
Π ij = ρp i p j m + T ij + V δ ij , (23)
T ij = h2 2m ∂A ∂x i ∂A ∂x j -A ∂ 2 A ∂x i ∂x j , (24)
∂ρ ∂t + 1 m ∂ (ρp j ) ∂x j = 0, (25)
and ρ = A 2 . ( 26
)
Though those equations are nonlinear and certainly more complex than the equations of classical quantum mechanics they have an attractive feature: that they can be generalized to special relativity without too much effort or mathematical complications.

Those equations have solutions where ∇ × p = 0 which can be obtained by solving Schrödinger equation's, but equations (22-26) are more general, because as already established, all solutions of Schrödinger's equation correspond to a case where ∇ × p = 0 in the classical limit, as follows from the relation p = ∇S, which is the foundation of the de Broglie-Bohm theory. As a consequence, the operational formalism of classical quantum mechanics, which is a linear theory, does not apply to this more general case.

In the context of the de Broglie-Bohm theory, the assumption that p = ∇S has been challenged by Einstein, considering a very simple example of a particle in a one-dimensional box where

V (x) = 0 if 0 < x < a ∞ if x ≤ 0 or x ≥ a
In this case the wave functions of the stationary states are

ψ n (x, t) = 2 a sin (k n x) e -iEnt/h , k n = nπ/a.
The nodes of the wave function ψ n (x, t) divide the interval (0, a) into n subintervals I j = ((n -1) • a/n, j • a/n). If we take A = ψ n (x, t we have to conclude that S n (x, t) = -E n t if x ∈ I j and j is odd E n t if x ∈ I j and j is even As a consequence ∂S ∂x = 0 and ẋ = 0 if x ∈ I j , meaning that a particle in a stationary state, in particular a particle in the base state, is simply at rest at some point in the interval (0, a). Similar remarks apply to the stationary states of a hydrogen atom, because those wave functions can be chosen real. In our view, this difficulty can be avoided by admitting that the term -∇ h2 2m ∆A A does not represent the gradient of a potential energy, but the gradient of an internal pressure, in the same way as, when we study the aerodynamics of a perfect gas, we make a distinction between the kinetic energy of the internal motion of an element of fluid, and the kinetic energy of its motion as a whole. However then we have to solve the problem that this term is singular at points where A = 0, which we have already addressed with the introduction of the stress tensor. Those arguments, in our opinion, strongly speak in favor of a hydrodynamic interpretation instead of the de Broglie-Bohm pilot wave theory.

Relativistic particle in an external electromagnetic field

The equations of motion of a relativistic charged particle in an external electromagnetic field can be obtained from the Lagrange function [6, pp. 45-46]

L = -mc 2 1 - v 2 c 2 + q c A • v -qA 0 , (27)
where (A 0 , A) is the four-potential. The linear momentum is

p = ∂L ∂v = mv 1 -v 2 c 2 + q c A ( 28
)
and the energy:

E = mc 2 1 -v 2 c 2 + qA 0 ( 29
)
From those expressions we see that

E -qA 0 c 2 -p - q c A 2 = m 2 c 2 ,
and, in consequence

H = c p - q c A 2 + m 2 c 2 + qA 0 ( 30
)
Considering again p = p(r), if p = ∇φ, we get

∂φ ∂t + c ∇φ - q c A 2 + m 2 c 2 + qA 0 = 0. ( 31
)
In four dimensional notation this equation can be writen as:

- ∂φ ∂x a - q c A a - ∂φ ∂x a - q c A a = m 2 c 2 , ( 32
)
which is consistent with the formula

p a = mcµ a = - ∂φ ∂x a - q c A a = E -qA 0 c , ∇φ - q c A = E -qA 0 c , p - q c A
(33) for the field of kinetic linear four-momentum. Therefore, in this particular case, the vorticity of the kinetic four-momentum (the four-velocity) is not zero, but proportional to Faraday's tensor:

∂p a ∂x b - ∂p b ∂x a = - q c F ab . (34)
In particular, the curl of the three-dimensional kinetic momentum is not zero, but proportional to the magnetic field.

In general, [START_REF] Chavoya | An explanation of the quantum mechanics of particles[END_REF] takes the form:

∂p ∂t + c∇ p - q c A 2 + m 2 c 2 + q∇A 0 = c p -q c A p -q c A 2 + m 2 c 2 × (∇ × p) .
(35) It is not difficult to prove that the field of velocities is given by

v = c p -q c A p -q c A 2 + m 2 c 2 . ( 36
)
By use of this in (35) we get

∂p ∂t + c∇ p - q c A 2 + m 2 c 2 + q∇A 0 = v × (∇ × p) .
Then, after a few transformations equation ( 35) is written as

∂p -q c A ∂t + c∇ p - q c A 2 + m 2 c 2 -v × ∇ × p - q c A = qE + q c v × H, (37) where
E = -∇A 0 - 1 c ∂A ∂t and H = ∇ × A.
Equation (37) can be further simplified to:

∂ p -q c A ∂t + (v • ∇) p - q c A = qE + q c v × H. (38)
Considering the components of the four-velocity field:

µ a =   1 1 -v 2 c 2 , v/c c 1 -v 2 c 2  
we can see that (38) is equivalent to the spatial part of the covariant equation

mcµ b ∂µ a ∂x b = q c F ab µ b (39)
where

F ab = ∂A b ∂x a - ∂A a ∂x b , (40)
is Faraday's tensor. By replacing (24) with

∂ ρµ b ∂x b = 0 (41)
and making

ρ = α 2 , (42)
τ ab Q = h2 2mc ∂α ∂x a ∂α ∂x b - ∂ 2 α x a x b , (43)
we get

mcρµ b ∂µ a ∂x b = q c ρµ b F ab - ∂τ ab Q ∂x b . (44)
In the classical limit of low speeds and macroscopic objects, equations (41-44) reduce to the equations of classical mechanics for this kind of problem. In the limit of low speeds and small masses they reduce to a set of equations including classical quantum mechanics as a special case, but more general, because they are not subject to condition (34).

Using equation ( 41), (44) can be written as

∂ ∂x b mcρµ a µ b + τ ab Q = q c ρµ b F ab . ( 45
)
In this equation we see that

j b = qρµ b
appears as a current density, which suggests us to consider Maxwell equations [6, p. 67 eq. (26.5) & p. 74 eq. (30.2)]:

∂F ab ∂x c + ∂F bc ∂x a + ∂F ca ∂x b = 0, (46)
and

∂F ab ∂x b = - 4π c j a (47)
Using (47) the right side of (45) can be written as

j b c F ab = - 1 4π F ab ∂F bc ∂x c . Then - 1 4π F ab ∂F bc ∂x c = - 1 4π ∂ ∂x c F bc F ab -F bc ∂F ab ∂x c . (48)
Using ( 46)

-F bc ∂F ab ∂x c = F bc ∂F bc ∂x a + ∂F ca ∂x b = 1 2 ∂ ∂x c δ a c F de F de + F bc ∂F ca ∂x b = 1 2 ∂ ∂x c δ a c F de F de + F bc ∂F ab ∂x c
(The last equality is obtained by interchanging the indexes b and c and using the fact that Faraday's tensor is antisymmetric.) Accordingly:

-F bc ∂F ab ∂x c = 1 4 ∂ ∂x c δ a c F de F de .
Equation ( 48) can be written as

- 1 4π F ab ∂F bc ∂x c = - 1 4π ∂ ∂x b -F ac F b c + 1 4 δ ab F cd F cd = - ∂τ ab E ∂x b
where

τ ab E = 1 4π -F ac F b c + 1 4 δ ab F cd F cd (49)
is the stress-energy tensor of the electromagnetic field [6, p.81 eq. 33.1].

Discussion

To summarize we include here a whole set of equations of a covariant hydrodynamic reformulation of quantum electrodynamics, or a possible self-consistent description of the interaction of matter and radiation:

∂ρµ a ∂x a = 0, (50)
(Expressing conservation of electric charge.)

ρ = α 2 , (51)
τ ab Q = h2 2mc ∂α ∂x a ∂α ∂x b - ∂ 2 α x a x b , (52)
∂F ab ∂x c + ∂F bc ∂x a + ∂F ca ∂x b = 0, (53)
(First pair of Maxwell equations.)

∂F ab ∂x b = - 4π c qρµ a (54)
(Second pair of Maxwell equations.)

τ ab E = 1 4π -F ac F b c + 1 4 δ ab F cd F cd (55) ∂ ∂x b mcρµ a µ b + τ ab Q + τ ab E = 0. (56)
(Expressing conservation of mass energy.)

Thought the concept a wave function is not mentioned here, nor probabilities, this won't be a causal theory, because the electrical current density, as well as the stress-energy tensor, depend on the density ρ. As we have stressed in the conclusions of [START_REF] Chavoya | An explanation of the quantum mechanics of particles[END_REF], the normalization condition that is imposed on the density ρ(x, t) in classical quantum mechanics implies to consider non overlapping regions of space as mutually exclusive events in the sense of the classical theory of probabilities. This means that a particle cannot be found inside two non overlapping regions of space at the same time, or simultaneously. The concept of simultaneity has been challenged by the special theory of relativity, based on the negative result of the Michelson-Morley experiment and other facts and, as a consequence, by Einstein's theory of gravitation. Accordingly, though massenergy and electric charge are conserved, the number of particles might not be conserved.

Also, a conspicuous feature of those equations is that spin does not play any role in them. This might appear as a strong reason to reject our ideas. However, as we have explained in a previous paper, there are also good reasons to believe that spin is not a fundamental property of matter [START_REF] Chavoya | An explanation of spin based on classical mechanics and electrodynamics[END_REF].

Finally, those equations and, in particular, the stress tensor in (56) can be used to describe in a self-consistent manner the interaction of matter, radiation, and gravitational field, including quantum effects. Of course, those equations refer to a single kind of particle, which means that they are, at most, one step towards a general field theory of matter, radiation, and gravitation.

By substitution into Einstein equation [6, p. 278] (our constant -8πG/c 3 is different because of the way we have defined the stress-energy tensor.) Thus, our hypotheses predict that quantum effects in the gravitational field will not be noticeable but for particles of very small mass (because of the presence of the factor λ C ) at very high energies for which α changes in distances of the order of Planck's length, as follows from the presence of the factor which is almost independent of the mass for microscopic particles. From this we draw the conclusion that quantum effects on the space-time metric are not to be expected but at very high energies, much higher than the energy that can be achieved in any existing accelerator.

As it has been mentioned in the introduction, we do not claim to have a complete theory of matter, radiation, and gravitation, because those equations refer to particles of a single kind.

3 .

 3 The path of the motion is obtained as the solution of the equation ẋ = (1/m) ∇S| x=x(t)

R ab - 1 2 g ab R = - 8πG c 3 c 3 τ

 33 mcρµ a µ b + τ E ab + τ Q ab Apart from the usual sources of the gravitational field, we have the term-8πG c 3 τ Q ab = -Q ab = -4πλ C l 2 P ∂α ∂x a ∂α ∂x b -α ∂ 2 α ∂x a ∂x b -Γ c ab ∂α ∂x c(57)where λ C = h mc is Compton's wave length andl p = hG c 3 is Planck's length.(58)

e p 2 P c 2 + m 2 c 4 = 1 e hc 5 G + m 2 c 4 ≈

 2254 Heuristically, the second condition requires the de Broglie wave-length to be of the order of Planck's length. The corresponding momentum isp P = h l pand the energy (Planck's Energy), is:E eV (m) = 1 1.221 × 10 19 G eV