EQUILIBRIUM CUSTOMER STRATEGIES AND SOCIAL-PROFIT MAXIMIZATION IN THE SINGLE SERVER CONSTANT RETRIAL QUEUE

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1 EQUILIBRIUM CUSTOMER STRATEGIES AND SOCIAL-PROFIT MAXIMIZATION IN THE SINGLE SERVER CONSTANT RETRIAL QUEUE ANTONIS ECONOMOU AND SPYRIDOULA KANTA Abstract. We consider the single server constant retrial queue with a Poisson arrival process and exponential service and retrial times. This system has not waiting space, so the customers that find the server busy are forced to abandon the system, but they can leave their contact details. Hence, after a service completion, the server seeks for a customer among those that have unsuccessfully applied for service but left their contact details, at a constant retrial rate. We assume that the arriving customers that find the server busy decide whether to leave their contact details or to balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine the customers behavior and we identify the Nash equilibrium joining strategies. We also study the corresponding social and profit maximization problems. We consider separately the observable case where the customers get informed about the number of customers waiting for service and the unobservable case where they do not receive this information. Several extensions of the model are also discussed. Keywords: queueing, constant retrials, balking, equilibrium strategies, pricing, social optimization, profit maximization, Nash equilibrium, partial information Published in 20 in Naval Research Logistics 58, The original publication is available with DOI: 0.002/nav Introduction In most studies in queueing theory, it is usually assumed that blocked customers (those who cannot get immediately service or waiting space to stay abandon the system for ever. However, in most applications, it is reasonable to assume that the customers retry for service after some random period of time. This is crucial for the accurate representation of a given system with a queueing model. Kosten (973 notes that any theoretical result that does not take into consideration this repetition effect should be considered suspect. Retrial queues have been introduced to model exactly this repetition effect. In the majority of papers in the retrial queueing literature, each blocked customer joins the socalled retrial orbit and becomes a source of repeated requests for service at rate ν, independently of the other customers. This is the classical retrial policy in which the total retrial rate when there are j customers in the orbit is jν. In contrast to this, there are some applications in computer and communication networks, where the time between two successive repeated attempts is controlled by some automatic mechanism and consequently the total retrial rate is, independently of the number j of customers in orbit. This constant retrial policy is also used to model systems in which the blocked customers leave their contact details when they find the server busy. Then, after a service completion, the server seeks for a customer at a constant (retrial rate, among those that have left their contact details. The literature on retrial queueing systems is already very extensive. For a recent account we point to the recent books of Falin and Templeton (997 and Artalejo and Gomez-Corral (2008 that summarize the main models and methods. The constant retrial policy was introduced in Fayolle (986. Subsequently several authors considered various complicated systems operating under this policy or its generalization, the so-called linear retrial policy (see e.g. Falin and Gomez-Corral (2000. However, the vast majority of papers on retrial queueing systems is devoted on performance evaluation and control problems that are solved using stochastic processes and dynamic programming techniques. There are very few papers that study this type of systems from an economic viewpoint, that is when the customers are allowed to take their own decisions (e.g. to join or balk,

2 2 ANTONIS ECONOMOU AND SPYRIDOULA KANTA to decide their individual retrial rate etc.. Indeed, if we impose a reward-cost structure and allow the customers to take their own decisions, the optimization problem becomes a game among the customers and at a first level the objective is to find the Nash equilibrium points. At a second level, the administrator of the system has to solve the social and profit maximization problems, taking into account the behavior of the customers. These ideas of a game-theoretic analysis of queueing systems began with the pioneering works of Naor (969 and Edelson and Hildebrand (975 who identified Nash equilibrium balking strategies for the observable and unobservable M/M/ queue respectively. This kind of analysis is particularly involved in the case of partial information, i.e. whenever the customers take their decisions without knowing the exact state of the system. Under partial information, the studies are still focused on classical models (see e.g. the analysis of the M/M/ queue with priorities in Hassin and Haviv (997, of the M/G/ queue in Haviv and Kerner (2007, of the M/M/ queue with setup times in Burnetas and Economou (2007 and of the M/M/ queue with server s breakdowns in Economou and Kanta (2008a. For a thorough summary of the state-of-the-art in this subarea of queueing and its applications see the monographs of Hassin and Haviv (2003, Stidham (2009 and the review papers of Altman et al. (2006 and Aksin et al. (2007. In the framework of retrial queues, this type of game-theoretic analysis has been done for some models with the classical retrial policy. In these studies, it is always assumed that the customers have only partial information of the state of the system, that is, they can observe the state of the server(s (busy-idle, but they cannot know how many customers are in the retrial orbit. Hence the equilibrium customer behavior literature in retrial queues deals with basically unobservable models. Kulkarni (983 and Elcan (994 derived the socially optimal and equilibrium retrial rates for the case of the single server Markovian retrial queue with the classical retrial policy, while Hassin and Haviv (996 extended the analysis in the case of general service times. To the best of our knowledge, there are no papers dealing with equilibrium customer strategies and/or the social and profit maximization problems in the class of retrial systems with the constant retrial policy. Nor do there exist studies for observable models in the class of retrial systems. In the present paper we aim to investigate these problems in the framework of the basic single server constant retrial queue. More concretely, we study separately two information cases: the unobservable case where the customers know only the state of the server (busy or idle and the observable case where they also get informed about the number of customers in the retrial orbit. Moreover, we perform some numerical experiments and discuss extensions of the theoretical results. The paper is organized as follows. In Section 2 we describe the dynamics of the model, the reward-cost structure and the decision assumptions for the customers. In Section 3 we determine Nash equilibrium, social and profit maximizing strategies for joining the retrial orbit in the unobservable case. In Section 4 we treat the corresponding observable case. Finally, in Section 5, we present several numerical results that show the effect of the model parameters on the behavior of the customers. We also demonstrate numerically the value of information for the customers (observable vs. unobservable model and discuss the qualitative implications of the results. Some further generalizations of the results are also discussed. 2. The model We consider a single server queueing system with no waiting space in which customers arrive according to a Poisson process at rate. The service times of the customers are assumed to be exponentially distributed random variables at rate. The customers that find the server busy upon arrival, they abandon the system but leave their contact details; hence we can think that they join a virtual retrial orbit or that they are registered in a server s waiting list. After finishing service, a customer leaves the system and the server seeks to serve a customer from the retrial orbit. The time required to find a customer from the retrial orbit is assumed exponentially distributed with rate. However, it is possible that during the seeking process, a new customer arrives. In this case, the server interrupts the seeking process and serves the new customer. We assume that the inter-arrival, service and seeking times are mutually independent.we will refer to this model as the single server constant retrial queue.

3 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE 3 We represent the state of the system at time t by a random vector (I(t, X(t, where I(t denotes the state of the server (0: idle, : busy and X(t is the number of customers in the orbit. It is obvious that the stochastic process {(I(t, X(t} is a continuous-time Markov chain with state space S = {0, } {0,, 2,...} and non-zero transition rates given by q (0,j(,j =, j = 0,, 2,... (2. q (,j(0,j =, j = 0,, 2,... (2.2 q (,j(,j+ =, j = 0,, 2,... (2.3 q (0,j(,j =, j =, 2, 3,.... (2.4 The corresponding transition rate diagram is shown in figure 2.. 0,0 0, 0,2 0, ,0,,2,3 Figure 2.: Transition rate diagram of the original model We are interested in the behavior of the customers when they can decide whether to join or balk upon their arrival. We assume that every customer receives a reward of R units for completing service. This reward quantifies his satisfaction and/or the added value of being served. Moreover, there exists a waiting cost of C units per time unit that is continuously accumulated from the time that the customer arrives at the system till the time he leaves after being served (this cost is accumulated both in orbit and in service. Customers are risk neutral in the sense that they want to maximize their expected net benefit (but they do not care for the corresponding variance. We assume that R > C. (2.5 If this condition fails to hold, then even the customers that find the server idle have not any incentive to enter the system, since their expected total waiting cost is equal or exceeds their reward from being served. So the condition (2.5 ensures that the customers who find the server idle do enter in the system. We consider separately two information cases. In the first case that we coin unobservable, we suppose that the customers are informed upon their arrival only about the state of the server. In the second case that we refer to as observable, we suppose that they also get informed about the number of customers that wait for service in the orbit. In both information cases, the customers that find an idle server will certainly enter to receive service because of the condition (2.5. The customers that find a busy server have to decide whether to leave their contact details and enter to the retrial orbit (entering customers or leave for ever (balking customers. We further assume that the decisions are irrevocable: retrials of balking customers and reneging of entering customers are not allowed. We can then think of this situation as a symmetric game played by the customers, since they are all indistinguishable. Denote the common set of strategies and the payoff function by S and F, respectively. More concretely let F (a, b be the payoff of a customer that selects strategy a when everyone else selects strategy b. A strategy s e is a (symmetric Nash equilibrium if F (s e, s e F (s, s e, for every s S. This means that it is a best response against itself, i.e., if all customers follow it, no one can benefit by changing it. A strategy s is said to weakly dominate strategy s 2 if F (s, s F (s 2, s, for every s S and for at least one s the inequality is strict. A strategy s is said to be weakly dominant if it weakly dominates all other strategies in S. We will study the problem of identifying the equilibrium strategies for the unobservable case in Section 3, while we consider the observable case in Section 4.

4 4 ANTONIS ECONOMOU AND SPYRIDOULA KANTA 3. The unobservable case As we have already noted, a customer that finds the server idle prefers to enter, because his reward exceeds the expected waiting cost due to the condition (2.5. In fact, the net benefit of such a customer is not influenced by the strategy of any other customer. Therefore, we concentrate our study on the behavior of the customers that find the server busy. The model is unobservable so a (mixed strategy of a customer is completely specified by the probability r of joining whenever he sees the server busy. To identify the equilibrium, social and profit maximizing strategies we should first investigate the stationary distribution of the system when all customers follow a given strategy r. Then the transition rates of the corresponding Markov chain are given by (2.-(2.4, where (2.3 has been replaced by The transition rate diagram is shown now in figure 3.. q (,j(,j+ = r, j = 0,, 2,.... (3. 0,0 0, 0,2 0, r r r,0,,2,3 r Figure 3.: Transition rate diagram of the unobservable case We have the following proposition. Proposition 3.. Consider the unobservable single server constant retrial queue, in which the customers enter with probability r whenever the server is busy and with probability whenever the server is idle. The system is stable if and only if ρ = r( + <. (3.2 In this case, the stationary probabilities of idle and busy server are given respectively by p 0 = p = r + ( r (3.3 + ( r. (3.4 Moreover, the expected (unconditional sojourn time in the system of an arriving customer and the expected (conditional sojourn time of a customer given that he sees upon arrival the server busy and decides to join the retrial orbit are given respectively by E[S] = E[S I = ] = ( + r ( + ( r( r( + ( r( + +. (3.6 Proof. Let (p i,j : (i, j S be the stationary distribution of the Markov chain {(I(t, X(t}. The balance equations for the stationary distribution are p 0,0 = p,0 (3.7 ( + p 0,j = p,j, j =, 2,... (3.8 rp,j = p 0,j+, j = 0,,.... (3.9

5 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE 5 Using (3.8-(3.9, we can express p 0,j+ in terms of p 0,j so these equations can be recursively solved and using the normalization equation we derive p 0,j = p,j = + ( r + ( δ j0 ( ρρj, j = 0,, 2,... (3.0 + ( r ( ρρj, j = 0,, 2,... (3. where ρ is given by (3.2 and δ ji is Kronecker s delta being if j = i and 0 otherwise. Summing (3.0 and (3. for all j = 0,,... yield (3.3 and (3.4 respectively. Because of the PASTA property the probability p arr that a customer finds the server busy upon arrival coincides with the stationary probability of a busy server p, so the total arrival rate in the retrial box is ret = rp arr 2 r = + ( r. (3.2 On the other hand the stationary mean number of customers in the retrial orbit is computed by (3.0-(3.: 2 r( + + E[X] = j(p 0,j + p,j = ( + ( r( r( +. (3.3 j= By applying Little s law in the retrial orbit we obtain the mean waiting time at the retrial orbit of a customer that enters to the retrial orbit as ret E[X] = ++ r(+. Summing the mean service time yields (3.6. For the mean sojourn time of a customer, E[S], using the PASTA property we have E[S] = p 0 + p (( r 0 + r E[S I = ] (3.4 which by substituting (3.3, (3.4 and (3.6 yields (3.5. In what follows we will assume that ( + <, (3.5 i.e. that the original system without customers decisions is stable. This implies obviously that the system is also stable under any strategy r that the customers may follow. This simplifies the presentation of the results. The results for the case where (3.5 does not hold can be easily concluded from this case as we comment later on. We are now in position to derive the equilibrium joining strategies for the customers. We have the following. Theorem 3.2. In the unobservable single server constant retrial queue, in which the conditions (2.5 and (3.5 hold, there exists a unique mixed equilibrium joining strategy enter the retrial orbit with probability r e whenever finding the server busy. The probability r e is given by where r e = 0, if R C t Le (3.6 r e = ( ( R ( + C, if t Le < R C < t Ue (3.7 r e =, if R C t Ue (3.8 t Le = (3.9 t Ue = ( +. (3.20

6 6 ANTONIS ECONOMOU AND SPYRIDOULA KANTA Proof. Suppose that all the customers follow a given joining strategy r. Then the system behaves in stationarity as described in Proposition 3.. So the expected net benefit of a tagged customer that finds the server busy and decides to join the retrial orbit is computed using (3.6 to be S e (r = R CE[S I = ] = R C ( + + r( + +. (3.2 When R C (, t Le], with t Le given by (3.9, we have that (3.2 is negative for all r [0, ] so the best response for the tagged customer that finds the server busy is not to enter. So in this case, balking is a dominant strategy and therefore the unique equilibrium point is r e = 0 and we have (3.6. When R C (t Le, t Ue, with t Le and t Ue given by (3.9 and (3.20, we can easily check that (3.2 has a unique root in (0, which is given by (3.7. This is the unique equilibrium point in this case. When R C [t Ue,, with t Ue given by (3.20, we conclude that (3.2 is positive for all r [0, ] so the best response for the tagged customer that finds the server busy is always to enter. So in this case, joining is a dominant strategy and therefore the unique equilibrium point is r e = and we have (3.8. Because of the monotonicity of the function S e (r, we have that whenever the joining probability r is smaller than r e then the unique best response is. For r = r e any strategy is a best response and for r > r e the unique best response is 0. This shows that the best response is a non-increasing function of the strategy, that is we have an Avoid-The-Crowd (ATC situation. We can now proceed to the problem of social optimization and profit maximization. We are interested in finding the joining probabilities r soc and r prof that maximize the social net benefit and the administrator s profit per time unit. For the social optimization problem the solution is summarized in the following theorem. Theorem 3.3. In the unobservable single server constant retrial queue, in which the conditions (2.5 and (3.5 hold, there exists a unique mixed joining strategy enter the retrial orbit with probability r soc whenever finding the server busy that maximizes the social net benefit per time unit. The probability r soc is given by where r soc = 0, if R C t Lsoc (3.22 r soc = + A ( A( +, if t Lsoc < R C < t Usoc (3.23 r soc =, if R C t Usoc (3.24 A = R + C C( + (3.25 t Lsoc = ( + + ( (3.26 ( ( + ( ( + t Usoc = ( ( + 2. (3.27 Proof. For a given joining strategy r for the customers who find the server busy, the system behaves in stationarity as described in Proposition 3.. Therefore the social net benefit per time unit is S soc (r = (rr CE[S(r], (3.28 where (r is the mean arrival rate for the customers that decide to enter the system and E[S(r] is the mean sojourn time of a customer of the system (irrespectively of his decision to enter or balk under the strategy r. Then, using (3.3-(3.4 and PASTA we obtain (r = p 0 + rp = + ( r. (3.29

7 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE 7 Moreover, the mean sojourn time E[S(r] of a customer in the system is given by (3.5, so (3.28 assumes the form (R + C S soc (r = + ( r C r( +. (3.30 We have that d dr S + ( r soc(r = 0 r( + = R + C (3.3 C( + + A and solving for r we obtain that the unique maximum of S soc (r is attained at ( A(+, where A is given by (3.25. Therefore r soc given by (3.23 is the social maximizing strategy given that it belongs to (0,. By solving the inequalities with respect to R C we have that this indeed happens if and only if t Lsoc < R C < t Usoc with t Lsoc and t Usoc given by (3.26 and (3.27 respectively. In case where R C t Lsoc we have that S soc (r is non-increasing in [0, ] so it takes its maximum at 0, while for R C t Usoc we have that S soc (r is non-decreasing in [0, ] so it takes its maximum at. Therefore we obtain (3.22-(3.24. We now consider the profit maximization problem. In this case we suppose that the administrator of the system imposes an entrance fee p for the customers with the objective of maximizing his own profit per time unit. By imposing this fee the reward for the customers changes from R to R p and therefore a new strategy r prof (p is their new equilibrium strategy. We are interested in determining the strategy r prof that maximizes the administrator s profit per time unit. We have the following Theorem 3.4. In the unobservable single server constant retrial queue, in which the conditions (2.5 and (3.5 hold, there exists a unique mixed joining strategy enter the retrial orbit with probability r prof whenever finding the server busy that maximizes the administrator s net reward per time unit. The probability r prof is given by where r prof = 0, if R C t Lprof (3.32 r prof = + B ( B( +, if t Lprof < R C < t Uprof (3.33 r prof =, if R C t Uprof (3.34 B = R + C + (3.35 t Lprof = + ( + 2 ( (3.36 t Uprof = + ( ( ( + 2. (3.37 Proof. For a given joining strategy r we define S prof (r to be the administrator s net profit per time unit, when the customers follow the strategy r to respond optimally to the price that he imposes. Let (r be the mean effective arrival rate when the customers follow strategy r and p(r be the entrance fee that the administrator imposes to induce the strategy r. Then we have S prof (r = (rp(r (3.38 where the arrival rate (r is given by (3.29. To determine p(r in terms of r, we consider a customer that arrives to the system when an entrance fee p is imposed. This customer reacts optimally according to Theorem 3.2 with R replaced by R p. Therefore we have to solve the equation ( + + R p = C r( + +, (3.39

8 8 ANTONIS ECONOMOU AND SPYRIDOULA KANTA from which we obtain p(r = R C ( + + C, r [0, ]. (3.40 r( + Therefore the function S prof (r assumes the form ( S prof (r = (rp(r = R C + C + ( r C ( + r( +. (3.4 We have that d dr S + ( r prof (r = 0 r( + = R + C + ( B and solving for r we obtain that the unique maximum of S prof (r is attained at ( B(+, where B is given by (3.35. Therefore r prof given by (3.33 is the profit maximizing strategy given that it belongs to (0,. The rest of the proof proceeds along the same lines with the proof of Theorem 3.3. Indeed, observe that the functions S soc (r, S prof (r given by (3.30 and (3.4 are both of the form c +( r +c 2 r(+ (c, c 2 being constants so their optimization is carried out almost identically. We now proceed to the comparison of the optimal joining probabilities for the three different aspects of the problem (individual, social and profit maximization. For the proof we again assume that the condition (3.5 holds. The result is valid in general but the proof needs to be slightly modified in case that (3.5 fails to hold. Theorem 3.5. The optimal joining probabilities r e, r soc and r prof are ordered as r prof r soc r e. (3.43 Proof. First we will prove that r soc r e. When the condition (3.5 holds, it is easily shown that t Le < t Lsoc < t Ue < t Usoc, where t Le, t Ue and t Lsoc, t Usoc are defined in Theorems 3.2 and 3.3 respectively. We consider the following cases: If R C t Le then according to (3.6 and (3.22, r soc = r e = 0. If t Le < R C t Lsoc then r e (0, given by (3.7 and r soc = 0. So the inequality r soc < r e holds. If t Lsoc < R C < t Ue then r e (0, given by (3.7 and r soc (0, given by (3.23. After some algebra we obtain that r soc < r e. If t Ue R C < t Usoc then r e = and r soc (0, given by (3.23 and obviously r soc < r e. Finally, if R C t Usoc then r e = r soc =. So, in any case the second inequality of (3.43 is valid. Figure 3.2 makes clear the various cases of the proof, where + denotes the value of the corresponding probability in (0, as given by (3.7 for the equilibrium probability and by (3.23 for the socially optimal one. (0,0 (0,+ (+,+ (+, (, (r soc,r e R t Le t Lsoc t Ue t Usoc C Figure 3.2: Socially optimal and equilibrium joining probabilities with respect to R C We now proceed with the proof of the first inequality of (3.43. Again, it can be easily seen that t Lsoc < t Lprof and t Usoc < t Uprof, with t Lsoc, t Usoc and t Lprof, t Uprof as defined in Theorems 3.3 and 3.4 respectively. However, unlike the comparison of r e and r soc where we have always t Lsoc < t Ue, the respective inequality t Lprof < t Usoc does not hold in general. So we consider two different cases according to whether t Lprof < t Usoc or t Lprof t Usoc. In both cases, the rest of the proof proceeds along the same lines with the proof of the second inequality of (3.43, i.e., we consider the various cases for the value of R C and we use the results of Theorems 3.3 and 3.4. Figure 3.3 clarifies our implications, where as before, + denotes the value of the corresponding

9 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE 9 probability in (0, as given by (3.23 for the socially optimal probability and by (3.33 for the profit maximizing one. (0,0 (0,+ (+,+ (+, (, (r prof,r soc R t Lsoc t Lprof t Usoc t Uprof C (t Lprof < t Usoc (0,0 (0,+ (0, (+, (, (r prof,r soc R t Lsoc t Usoc t Lprof t Uprof C (t Lprof t Usoc Figure 3.3: Profit maximizing and socially optimal joining probabilities with respect to R C The above result agrees with similar results concerning the economic analysis of other queueing models: Whenever each individual uses a common resource to maximize his own utility, then in equilibrium the customers make excessive use of the resource (see e.g. Hardin (968. In other words, individual optimization leads to systems more congested than what is socially desirable. This happens because the customers try to maximize their own net profit without caring for the negative externalities that they impose on the other customers (that arrive later at the system. 4. The observable case We now turn our attention to the observable case where the customers get informed not only about the state of the server, but also about the exact number of customers in the orbit in front of them. Obviously, this information is useful only for the customers that find the server busy. More concretely, this additional information matters for these customers, since after every service completion the server begins a seeking process. If no customer arrives during the seeking period, the server selects the first customer of the orbit to serve. That is, the customers of the orbit are served in a FCFS basis. Hence knowing their position in the orbit, they can assess more precisely whether it is preferable or not to enter the system. On the other hand this information is useless for those that find the server idle, since they begin to be served immediately. As in the unobservable case, the expected sojourn time of these customers is, not dependent on the arrival rate, nor on the strategies of the others. Consider now, a tagged customer that finds the server busy upon arrival. This customer s mean overall sojourn time in the system is not influenced by the join/balking behavior of the other customers that find the server busy. Indeed, assuming that the tagged customer joins the system at the jth position of the orbit, the future customers that find the server busy will occupy the positions j +, j behind him in the orbit. So his sojourn time does not depend on their decisions. However, it depends on the arrival rate, because of the future customers that arrive when the server is idle and join the system (under the condition (2.5. To see this more clearly, we suppose that the customers of such a system follow a general strategy, which prescribes that they enter to the system when they find the server idle (because of (2.5 and that they join the retrial orbit with probability s j when they find the server busy and j customers in the orbit. Let T (i, j be the mean (residual sojourn time of a tagged customer that he is at the jth position in the orbit, given that the server is at state i, when all the customers follow the general strategy. By a first step argument we have that T (i, j satisfies the linear system of equations T (, 0 = (4. T (, j = s j + + T (0, j = + + s j s j + T (, j + T (0, j, j =, 2,... s j + (4.2 T (, j + T (, j, j =, 2, (4.3

10 0 ANTONIS ECONOMOU AND SPYRIDOULA KANTA Note that T (, 0 denotes the mean residual service time of a customer that is currently served and it is equal to. Let us consider a customer in the jth position of the orbit with idle server. Then, he has to wait for an exponentially distributed time with rate + for the next event to occur, which is a new arrival or the end of the seeking period of the server. With probability + a new customer arrives at the system and since the server is free, he begins immediately his service, so the tagged customer remains in the jth position of the orbit. On the other hand, with probability + the seeking period of the server is completed and the server chooses the first customer in the orbit to serve, so he becomes busy and the tagged customer moves to the position j. Hence we deduce equation (4.3. With a similar argument equation (4.2 is deduced. Solving (4.2 with respect to T (0, j and substituting in (4.3 we obtain that T (, j = + + which is solved iteratively. Taking into account (4., we obtain that Substituting in (4.3 yields T (, j = j T (, j, j =, 2,... (4.4 +, j = 0,,.... (4.5 T (0, j = j + +, j =, 2,.... (4.6 Therefore, we see that indeed the mean conditional (residual sojourn times T (i, j of a tagged customer given the state (i, j of the system do not depend on the general strategy specified by the probabilities s, s 2,... of the others. To avoid trivialities, we assume that R CT (, > 0, otherwise a customer that finds the server busy does not join the orbit even if the orbit is empty. This condition is written equivalently as R C > and is stronger than the condition (2.5 (in fact it is equivalent to t Le < R C with t Le given by (3.9 - a condition that ensures that the equilibrium joining probability is positive in the unobservable case. We are now in position to identify the equilibrium strategies of the customers. Theorem 4.. In the observable single server constant retrial queue, in which the condition (4.7 holds, there exists the threshold equilibrium joining strategy enter the retrial orbit if there are at most n e customers in the orbit, whenever finding the server busy. The threshold n e is given from (4.7 n e = x e, (4.8 where ( R x e = + + C. (4.9 This strategy is the unique equilibrium strategy among all possible strategies, provided that a customer decides to enter, whenever he is indifferent between joining and balking. Proof. Consider a tagged customer that finds upon arrival the system at state (, j. If he decides to join, he goes directly to the orbit where he is assigned at the (j + th position. Then his expected profit is S e (j = R CT (, j + which using (4.5 assumes the form ( S e (j = R C (j (4.0 The customer prefers to enter if S e (j > 0, he is indifferent between joining and balking if S e (j = 0 and prefers to balk if S e (j < 0. By solving S e (j 0 for j, we obtain that the arriving customer prefers to enter if and only if j n e, where n e is given by (4.8-(4.9. Since this strategy is preferable independently of the strategy that the other customers follow, we have immediately that it is a weakly dominant strategy. Its uniqueness as an equilibrium among all possible strategies is obvious, if we have that a customer decides to enter whenever he is indifferent between joining and balking. Indeed, in case of x e being an integer and hence n e = x e, we have that S(n e = 0 and

11 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE any mixed threshold strategy which prescribes to enter if n < n e, to balk if n > n e and randomizes for n = n e is an equilibrium strategy. Therefore, in this case we need to assume that the customer decides to enter, whenever he is indifferent between joining and balking, to ensure the uniqueness of the equilibrium strategy. We will now proceed to the problem of social optimization and profit maximization. We are interested in finding the thresholds n soc and n prof that maximize the social net reward and the administrator s profit per time unit. To this end we need first to determine the stationary behavior of the system, when the customers follow a threshold strategy. More concretely, let assume that all customers that find the server busy follow the same threshold strategy, i.e. they join the system as long as the number of customers in orbit is less than or equal to n and balk otherwise. Hence, the maximum number of customers that can be accumulated in the orbit is n +. It is obvious that the stochastic process {(I(t, X(t} is a continuous-time Markov chain with state space S = {0, } {0,, 2,..., n + } and its transition rate diagram is given in figure 4.. 0,0 0, 0, ,0,,2 0,n ,n+ 4,n,n+ Figure 4.: Transition rate diagram of the observable case The stationary distribution of the system, (p i,j : (i, j S, is given in the following proposition. Proposition 4.2. Consider the observable single server retrial queue, in which the customers follow the threshold-n strategy, i.e. they join the orbit as long as the number of customers in it is less than or equal to n, whenever the server is busy and enter without restriction, whenever the server is idle. The system is stable and the stationary distribution is given by p 0,0 = A(n 2 (4. p 0,j = A(nρ j, j =, 2,..., n + (4.2 p,j = A(n ρj, j = 0,,..., n + (4.3 where ( + ρ = (4.4 ( A(n = 2 + ( + ρ + + ρn + ( + ρ + + ρn+. (4.5 Proof. The stationary distribution is obtained as the unique normalized solution of the system of the balance equations p 0,0 = p,0 (4.6 ( + p 0,j = p,j, j =, 2,..., n + (4.7 p,j = p 0,j+, j = 0,,..., n (4.8 p,n+ = p,n + p 0,n+. (4.9 The computations for the stationary distribution are done along the same lines with the corresponding computations in the unobservable case as they are presented in the proof of Proposition 3.. More concretely, by solving (4.7 for p,j and substituting in (4.8 we obtain that p 0,j = ρp 0,j j = 2, 3,..., n + which can be solved recursively and we can easily conclude with (4.2. Substituting (4.2 in (4.7 we obtain (4.3 for j =, 2,..., n +. Using (4.8 for j = 0 and (4.6 we obtain (4.3 for j = 0 and (4.. The normalization parameter A(n can then be

12 2 ANTONIS ECONOMOU AND SPYRIDOULA KANTA computed by the normalization equation. Note that for n = we have the threshold-( strategy which corresponds to the strategy where all customers that find the server busy balk (even when the orbit is empty. Note that in this case the equations (4.-(4.3 assume the form p 0,0 = p,0 = +, +, which agree with the formulas for the M/M// queue. Indeed, in this case the Markov chain {(I(t, X(t} has state space S = {(0, 0, (, 0} and it reduces to the M/M// queue. In what follows, we assume that ρ. This assumption facilitates the exposition, since the geometric sums of type j ρj that arise throughout the analysis (see e.g. (4.5 can be handled in a unified manner. The case ρ = can be treated similarly. We can now determine the threshold n soc that maximizes the social net profit per time unit. The result is given in the following theorem. Theorem 4.3. In the observable single server constant retrial queue, in which the condition (4.7 holds, let g(x = + ρ (x + + ((x + ρ ρx+. ( ( + ( ρ ρ If g(0 > x e, then the strategy that maximizes the social net benefit per time unit prescribes that all customers who find a busy server should balk. Otherwise there exists a unique threshold joining strategy enter the retrial orbit if there are at most n soc customers in the orbit, whenever finding the server busy that maximizes the social net benefit per time unit. The threshold n soc is given from n soc = x soc, (4.2 where x soc is the unique nonnegative solution of the equation g(x = x e. Proof. When all customers follow the same threshold-n strategy, i.e. they join as long as the number of customers in orbit is less than or equal to n, the effective arrival rate at the system is (n = ( p,n+ (n, where by p i,j (n we refer to the stationary probabilities that correspond to the threshold-n system. Using the results of Proposition 4.2, (n assumes the form ( (n = A(n ρ n+ = A(nρ n+. (4.22 The expected number of customers in system, E[Q(n], can be computed using Little s law as E[Q(n] = (ne[s (n], where S (n denotes the sojourn time of a customer that enters in the system with the threshold-n policy. By conditioning on the state found upon arrival by a customer in the system with the threshold-n policy, we obtain E[S (n] = n+ j=0 p 0,j (n p,n+ (n n + j=0 = + p,n+ (n + + ( p,j (n p,n+ (n (j + n (j + p,j (n. (4.23 j=0

13 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE 3 Using (4.22 and (4.23 we obtain that the expected social net benefit per time unit that corresponds to the threshold-n model is given as S soc (n = (nr C (n n (j + p,j (n ( = (n R C C + + = C + + = C + + (n + + (n(x e + j=0 n (j + p,j (n j=0 ( R C n (j + p,j (n. j=0 n (j + p,j (n. (4.24 We will now show that S soc (n is unimodal and attains its maximum at n soc given in the statement of the theorem. To this end we study the increments S soc (n S soc (n. After a bit of algebraic manipulation we can easily show that where S soc (n S soc (n 0 n n (x e + ( (n (n (j + p,j (n (j + p,j (n j=0 g(n x e, (4.25 j=0 g(n = g (n, (4.26 g 2 (n n n g (n = (j + p,j (n (j + p,j (n, j=0 g 2 (n = (n (n. The functions g (n and g 2 (n can be considerably simplified. After some straightforward algebra we obtain A(nA(n ρn ( g (n = (n + ( ρ 2 2 ( ρ 2 ρ + 2 ρ n+ + ρ n+2 (4.27 g 2 (n = 2 ρ n 2 A(nA(n. (4.28 Plugging (4.27 and (4.28 in (4.26, we obtain that g(n is given by (4.20. To prove now the unimodality of S soc (n, it is now sufficient to prove that g(x is increasing. To this end, define the function f : [0, R with f(x = g(x x. (4.29 We have d dx f(x = ρ ρ + ( + ρρx+2 ln ρ ( + ( ρ 2 0, x 0. (4.30 Therefore f(x is increasing and so is g(x = f(x + x. Moreover, note that lim x g(x =. Therefore either g(0 > x e in which case g(n > x e, n = 0,,... or there exists a unique n soc such that g(n soc x e < g(n soc +. (4.3 In the former case, in light of (4.25, we have that S soc (n is decreasing and so it is social optimum for the customers that find a busy server to balk. In the latter case, in light of (4.25, we have that S soc (n S soc (n 0, for n n soc while S soc (n S soc (n < 0, for n > n soc. The j=0 j=0

14 4 ANTONIS ECONOMOU AND SPYRIDOULA KANTA maximum of S soc (n occurs in n soc = x soc with x soc being the unique solution of the equation g(x = x e. We now consider the profit maximization problem. By imposing a nonnegative admission fee p, the administrator can force the customers to adopt any desired threshold smaller than n e. Although it has no practical sense, it is convenient to allow also negative values of p, so that the administrator can force the customers to adopt any desired joining threshold. We are interested in finding the threshold n prof that maximizes the administrator s profit. We have the following theorem. Theorem 4.4. In the observable single server constant retrial queue, in which the condition (4.7 holds, let h(x = x + ( ρx+ ( +ρ + ρx+2 ρ x ( ρ 2. (4.32 If h(0 > x e, then the strategy that maximizes the administrator s profit per time unit prescribes that all customers who find a busy server should balk. Otherwise there exists a unique threshold joining strategy enter the retrial orbit if there are at most n prof customers in the orbit, whenever finding the server busy that maximizes the administrator s profit per time unit. The threshold n prof is given from n prof = x prof, (4.33 where x prof is the unique nonnegative solution of the equation h(x = x e. Proof. The administrator s profit per time unit is S prof (n = (np(n where p(n is the fee that he imposes on the customers to force them to follow a certain threshold n. This is obtained by (4.9 replacing R with R p. Therefore we have to solve ( + + R p = C + (n + (4.34 from which we obtain ( p(n = R C (n We conclude that the administrator s profit per time unit assumes the form ( ( S prof (n = (n R C (n (4.35 = (nc + + (x e n. (4.36 We will now prove that the function S prof (n is unimodal. Using (4.36, we can easily show that S prof (n S prof (n (n n + (n (n x e h(n x e, (4.37 where h(n = n + (n (n (n. After some algebraic manipulation, taking into account (4.22 and (4.5, the function h(n assumes the form given in (4.32. We can then check routinely that h(x is an increasing function and lim x h(x =. Therefore either h(0 > x e in which case h(n > x e, n = 0,,... or there exists a unique n prof such that h(n prof x e < h(n prof +. (4.38 In the former case, in light of (4.37, we have that S prof (n is decreasing and so the administrator s profit is maximized when the customers that find a busy server balk. In the latter case, in light of (4.37, we have that S prof (n S prof (n, for n n prof while S prof (n < S prof (n, for n > n prof. The maximum of S prof (n occurs in n prof = x prof with x prof being the unique solution of the equation h(x = x e.

15 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE 5 The optimal thresholds n e, n soc and n prof are always in a certain order. In particular, we have the following theorem that is the analogue of Theorem 3.5 for the observable case. Theorem 4.5. The optimal joining thresholds n e, n soc and n prof are ordered as n prof n soc n e. (4.39 Proof. We will first prove that n soc n e. Note that f(x given by (4.29 is increasing because of (4.30. Moreover f(0 = g(0 > 0, so f(x soc 0, i.e. g(x soc x soc 0. But g(x soc = x e and we obtain x soc x e. Taking floors yields n soc n e. To prove that n prof n soc, note first that S prof (n < 0 for n > x e because of (4.36, so we have immediately n prof n e. We have also established that n soc n e. Therefore in the case where n soc = n e, we have immediately n prof n soc. Suppose that n soc < n e. It suffices to show that for integers n such that n soc n n e, we have that S prof (n + S prof (n, i.e. S prof (n is decreasing in {n soc, n soc +,..., n e }. From the unimodality of the function S prof (n that was established in the proof of Theorem 4.4, we will then have that the maximum of S prof (n, namely n prof, occurs for some n n soc and we will obtain n prof n soc. From the characterization of n soc given in (4.3 and taking into account that g(x is increasing (see (4.29-(4.30, we have that g(n + g(n soc + > x e, n n soc. (4.40 To prove that S prof (n is decreasing in {n soc, n soc +,..., n e }, we have to prove in light of (4.37 that h(n + x e, n soc n < n e. (4.4 Therefore, it suffices to prove that h(n g(n, i.e. that n + ( ρn+ ( +ρ + ρn+2 ρ n ( ρ 2 + ρ (n ( + ( ρ This reduces after some simplification to prove ρn+ (n + ρ ( ρ 2 ρ n+ In the case ρ <, (4.43 is written equivalently as n + ρn+ ( ρρ n+ Note also that in this case we have So, to prove (4.44 it suffices to prove that { ( +ρ + ρn+2 + { ( +ρ + ρn+2 + ((n + ρ ρn+ ρ. (4.42 } ( + ρ ρ n. (4.43 } ( + ρ ρ n. (4.44 ρ n+ n+ ( ρρ n+ = ( j n +. (4.45 ρ j= ( +ρ + ρn+2 + ( + ρ ρ n. (4.46 But, after some simplification, this is seen to be equivalent to ρ + ( + ρ ρ + ρn+2 >, (4.47 which clearly holds for ρ <. In the case ρ >, the same line of argument with all the inequalities reversed, shows that (4.43 is again valid. Therefore the relationship n prof n soc is valid in all cases.

16 6 ANTONIS ECONOMOU AND SPYRIDOULA KANTA 5. Discussion: Numerical results - Theoretical extensions 5.. Numerical results. In the unobservable case, closed forms for all the three joining probabilities (equilibrium, social and profit maximizing have been obtained in Section 3 as they are given in Theorems 3.2, 3.3 and 3.4. Hence, it is quite easy to see how each probability behaves, when keeping all but one parameters fixed and letting the one vary. Therefore, in this section, we focus on gaining qualitative insight for the observable case. Since, for this case, we have only implicit formulas for the social and profit maximizing thresholds in terms of floors of the roots of the equations g(x = x e and h(x = x e, we report the results of some numerical experiments that help us to understand the behavior of the three thresholds with respect to the parameters of the system. Moreover, we comment on the effect of the information available to the customers on the welfare, by making a comparison between the unobservable and the observable cases. In all applications we consider the waiting cost C per time unit and the service rate to be equal to. This can be always assumed after an appropriate re-scaling of reward/cost and time units. More concretely, in figure 5. the equilibrium, social and profit maximizing thresholds are presented, as the reward R increases for = 0.5 and = 5. We observe that all three thresholds are non-decreasing ladder functions of R. Regarding n e, by (4.9, we note that it increases in a linear scale with R. The ladder form can be explained by solving (4.9 for R C and hence we obtain that when R C [T (, n, T (, n +, n = 0,,... then n e = n, where T (, n is given in (4.5. On the other hand, the social and profit maximizing thresholds increase more slowly with respect to R. Furthermore, the order n prof n soc n e is illustrated, a result that is also valid in many other observable models. What the customers ignore at the time they make their decisions are the negative externalities that they impose on the other customers. Hence, individual optimization leads to more customers in orbit than it is socially desirable. Figure 5.2 shows the equilibrium, social optimizing and profit maximizing thresholds with respect to for R = 45 and =.2. Note that the equilibrium threshold is a non-increasing function of (it is also obvious by (4.9 and finally converges to 0. The social and profit thresholds are unimodal functions of and they become ultimately 0 as well for large values of. Recall that when the threshold is equal to zero, then only the customers that find the server idle join the system. Note also that the behavior of the thresholds as functions of the reward and the arrival rate as it appears in figures 5. and 5.2, is quite common in observable models (see for example Economou and Kanta (2008b when the joining - balking behavior of the customers is studied. Finally, a non-decreasing concave behavior of all thresholds with respect to the retrial rate of the server is shown in figure 5.3 in a scenario with R = 30 and = 0.9. As the retrial rate of the server increases the more willing are the customers that find him busy to join the system. Specifically for n e we can conclude by (4.9 that the threshold converges to R C as tends to infinity, a result that agrees with Naor (969 for the observable M/M/ queue. We also note that it is intuitively plausible that the effect of an increase in the retrial rate is stronger when is small compared to the external arrival rate. This justifies the concavity of the thresholds as functions of the retrial rate. Figures 5.4, 5.5 and 5.6 depict the social net benefit per time unit, when customers follow the corresponding equilibrium strategies in the observable and the unobservable case with respect to the parameters of the system, R, and respectively. More concretely, figure 5.4 shows that both functions exhibit a non-decreasing behavior as the reward R increases, in a scenario with = 0.4 and = 0.5. The other two numerical examples are performed for R = 7 and = 0.5 in figure 5.5 and R = 5 and = 0.5 in figure 5.6. Looking closely in figure 5.5, we notice that the social benefit, when the customers follow their equilibrium threshold strategy in the observable case, is a non-decreasing function of the arrival rate, even though the corresponding strategy is a non-increasing function of, as we have seen in graph 5.2. For the unobservable case, in the same graph, we observe a change in the convexity of the social benefit function under the equilibrium mixed strategy followed by the customers. This behavior can be explained by realizing that an increase in implies two competing opposite effects. On the one hand, when the congestion of the system is very low, the orbit is almost always empty so an increase in increases the reward per time unit, without increasing substantially the total waiting costs. But after a point, the customers continue to join the system, thinking individually,

17 JOINING STRATEGIES IN THE CONSTANT RETRIAL QUEUE 7 but the negative externalities (waiting costs for the customers in orbit begin to outrun the positive effect of the accumulated rewards. Therefore, for small values of the concave part that we have just described corresponds to equilibrium joining probability equal to one. On the other hand, as increases further, a new effect emerges as the equilibrium probability decreases from to 0 as it can be seen by (3.7. Then the function begins to increase rapidly and as tends to infinity, t Le goes to infinity too. Hence we have always the first case of Theorem 3.2, i.e. the joining probability is zero and the social benefit increases with respect to and it converges to R C (by (3.28. The limit R C corresponds to a system where only customers that find the server idle join the system, i.e. to an M/M// system with infinite arrival rate and service rate. Such a system has throughput and it is always busy so the net social reward per time unit is R C. Indeed, as increases, customers arrive more rapidly. Hence, the majority of arriving customers find the system busy. In this case, they are not willing to join the retrial orbit, since when the server will become empty, an external customer will arrive during the seeking process with high probability. Thus the customers in the orbit accumulate enormous waiting costs so the customers finding the server busy do not join the system. This shows why we have practically an M/M// system, and therefore the larger the arrival rate, the higher the reward rate per time unit. Regarding the value of information available to the customers and the benefit for the society, the situation is mixed. In the figures 5.4, 5.5 and 5.6 we have seen some numerical scenarios, where the unobservable case is preferable for some intervals of values for R, and while the observable case is preferable for other intervals of the parameters. In general, it seems that the additional information (the number of customers in orbit is beneficial for the society, when customers behave individually, for high values of R Linear retrial rates. In this subsection we consider the more general model proposed by Artalejo and Gomez (997, where the customers perform independently retrials at rate ν when they are in orbit, while the server seeks for a customer from the orbit at rate whenever he is idle. Then, the overall rate that a customer moves from the orbit to service is + jν, when the orbit size is j. We assume that the system is unobservable and customers that find the server idle always join under condition (2.5, while those that find him busy join with probability r. The transition rates of the corresponding Markov Chain {(I(t, X(t} are given by q (0,j(,j =, j = 0,, 2,... (5. q (,j(0,j =, j = 0,, 2,... (5.2 q (,j(,j+ = r, j = 0,, 2,... (5.3 q (0,j(,j = + jν, j =, 2, 3,.... (5.4 The corresponding transition rate diagram is shown in figure , ν 0, ν ,2 0, ,0 r, r,2 r,3 r ν Figure 5.7: Transition rate diagram of the unobservable case with linear retrial rates We introduce some notation. Let 2 F (a, b; c; z be the hypergeometric series given by 2F (a, b; c; z = where (x n is the Pochhammer symbol defined as n=0 (a n (b n (c n z n n!, (5.5 (x n = {, for n = 0 x(x + (x + n, for n. (5.6