A Discrete-Time Geo/G/1 Retrial Queue with General Retrial Times

Transcription

1 Queueing Systems 48, 5 21, Kluwer Academic Publishers. Manufactured in The Netherlands. A Discrete-Time Geo/G/1 Retrial Queue with General Retrial Times IVAN ATENCIA iatencia@ctima.uma.es Departamento de Matemática Aplicada, E.T.S.I. Telecomunicación, Universidad de Málaga, Campus de Teatinos, Málaga, Spain PILAR MORENO mpmornav@dee.upo.es Departamento de Economía y Empresa, Facultad de Ciencias Empresariales, Universidad Pablo de Olavide, Ctra. de Utrera, km. 1, Sevilla, Spain Received 21 November 2002; Revised 25 November 2003 Abstract. We consider a discrete-time Geo/G/1 retrial queue in which the retrial time has a general distribution and the server, after each service completion, begins a process of search in order to find the following customer to be served. We study the Markov chain underlying the considered queueing system and its ergodicity condition. We find the generating function of the number of customers in the orbit and in the system. We derive the stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions for our queueing system and its corresponding standard system. Also, we develop recursive formulae for calculating the steady-state distribution of the orbit and system sies. Besides, we prove that the M/G/1 retrial queue with general retrial times can be approximated by our corresponding discrete-time system. Finally, we give numerical examples to illustrate the effect of the parameters on several performance characteristics. Keywords: discrete-time model, general retrial times, Markov chain, recursive formulae, stochastic decomposition AMS subject classification: 60K25 1. Introduction There is a great potential for using the discrete-time queues in the performance analyses of computer and communication networks. The discrete-time queueing system has been found to be more appropriate in modeling computer and telecommunication systems because the basic time unit in these systems is a binary code. Moreover, the discretetime system can be used to approximate the continuous system in practice. The earliest work on discrete-time queue was presented by Meisling [13]. Recently, due to the fast progress of computer and telecommunication network technologies, the discrete-time queueing models have received more attention from queueing researchers. In last years, a number of queueing models have been analysed in discrete-time, details of which may be found in recent books [5,10,15,17].

2 6 ATENCIA AND MORENO Queueing systems with repeated attempts are characteried by the fact that a customer finding the server busy upon arrival must leave the service area and repeat his request for service after some random time. Between trials, the blocked customer joins a group of unsatisfied customers called orbit. Retrial queues have been widely used to model many practical problems in telephone switching systems, telecommunication networks and computers competing to gain service from a central processing unit. For a detailed review of the main results and the literature on this topic the reader is referred to [1,2,7]. In the past, the study of the retrial queues has been focused on the continuous case, but recently Yang and Li [18] have extended the study to discrete-time systems. Nevertheless, there are many papers on continuous-time retrial queues, but very little is known about discrete-time retrial queues. In all models of discrete-time retrial queues (see [4,6,11,12,16,18]), the time between retrials for any customer is assumed to be geometrically distributed. Our objective here is to extend the general retrial policy to the discrete-time retrial queues. The inherent difficulty with non-geometrical retrial times stems from the fact that the model must, in some way, keep track of the elapsed retrial time for each of possibly a very large number of customers. We avoid this problem by considering a Geo/G/1 retrial queue with general retrial times where only one customer may attempt retrials from orbit. Thus, the retrial discipline does not depend on the orbit sie. An important characteristic of the general retrial times policy is that we always obtain analytical solutions in terms of closed-form expressions. The general retrial times policy arises naturally in problems where the server is required to search for customers (see [14] in continuous-time), that is, this policy is related to many service systems where, after each service completion, the processor will spend a random amount of time in order to find the next item to be processed. The rest of the paper is organied as follows. In the next section, we give the model description of the considered queueing system. In section 3, we study the Markov chain and the stability condition of the system. Also, the distribution of the number of customers in the orbit and in the system is obtained. Besides, we obtain several performance measures of the system. In section 4, we find the stochastic decomposition property on the steady-state system sie and as an application we provide upper and lower estimates for the distance between the steady-state distributions for our queueing system and its corresponding standard system. In section 5 we derive a recursive algorithm for computing the steady-state probabilities of the numbers of customers in the orbit and in the system. In section 6, we analyse the relation between the continuous-time system and the discrete-time system. Finally, we discuss numerical results in section Description of the queueing system Let us consider a discrete-time single server retrial queue where the time axis is segmented into a sequence of equal time intervals (called slots). Further, let the time axis be marked by 0, 1,..., m,... It is assumed that all queueing activities (arrivals, departures and retrials) take place around the slot boundaries. For mathematical clarity,

3 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE 7 we assume that a potential departure occurs in the interval (m,m), and a potential arrival or retrial occurs in the interval (m, m + ); that is, arrivals and retrials occur at the moment immediately after the slot boundaries and the departures occur at the moment immediately before the slot boundaries. Customers arrive according to a geometrical arrival process with rate p, i.e., p is the probability that an arrival occurs in a slot. If an arriving customer finds the server free, he immediately begins his service. Otherwise, the customer leaves the service area and enters a group of blocked customers called orbit in accordance with an FCFS discipline. We will assume that only the customer at the head of the orbit is allowed for access to the server. It is always supposed that retrials and services can be started only at slot boundaries and their durations are integral multiples of a slot duration. Successive interretrial times of any customer are governed by an arbitrary distribution {a i } i=0 with generating function A(x) = i=0 a i x i. Service times are independent and identically distributed with general distribution {s i } i=1, generating function S(x) = i=1 s ix i and nth factorial moments β n. After service completion, the served customer leaves the system forever and will have no further effect on the system. In order to avoid trivial cases, we assume 0 <p<1. The traffic intensity is given by ρ = pβ The Markov chain At time m + (the instant immediately after time slot m), the system can be described by the process Y m = (C m,ξ 0,m,ξ 1,m,N m ), where C m represents the server state (0 or 1 according to the server is free or busy, respectively) and N m is the number of customers in the retrial group. If C m = 0and N m > 0, ξ 0,m denotes the remaining retrial time. If C m = 1, ξ 1,m corresponds to the remaining service time of the customer being served. It can be shown that {Y m,m N} is the Markov chain of our queueing system, whose states space is { } (0, 0); (0,i,k): i 1, k 1; (1,i,k): i 1, k 0. Our objective is to find the stationary distribution π 0,0 = lim P [C m = 0, N m = 0], m π 0,i,k = lim P [C m = 0, ξ 0,m = i, N m = k]; i 1, k 1, m π 1,i,k = lim P [C m = 1, ξ 1,m = i, N m = k]; i 1, k 0, m of the Markov chain {Y m,m N}. The one-step transition probabilities p yy = P [Y m+1 = y Y m = y] are given by the formulae p (0,0)(0,0) = p, p (1,1,0)(0,0) = p,

4 8 ATENCIA AND MORENO if i 1, k 1 p (0,i+1,k)(0,i,k) = p, p (1,1,k)(0,i,k) = pa i, if i 1, k 0 p (0,0)(1,i,k) = ps i, k = 0, p (0,1,k+1)(1,i,k) = ps i, p (0,j,k)(1,i,k) = ps i, j 1, k 1, p (1,1,k)(1,i,k) = ps i, p (1,1,k+1)(1,i,k) = pa 0 s i, p (1,i+1,k 1)(1,i,k) = p, k 1, p (1,i+1,k)(1,i,k) = p, where p = 1 p. The Kolmogorov equations for the stationary distribution are π 0,0 = pπ 0,0 + pπ 1,1,0, (1) π 0,i,k = pπ 0,i+1,k + pa i π 1,1,k ; i 1, k 1, (2) π 1,i,k = δ 0k ps i π 0,0 + ps i π 0,1,k+1 + (1 δ 0k )ps i π 0,j,k + ps i π 1,1,k + pa 0 s i π 1,1,k+1 + (1 δ 0k )pπ 1,i+1,k 1 + pπ 1,i+1,k ; i 1, k 0, (3) and the normaliation condition is π 0,0 + π 0,i,k + π 1,i,k = 1. i=1 k=1 i=1 k=0 In order to solve (1) (3) we introduce the generating functions ϕ 0 (x, ) = π 0,i,k x i k, ϕ 1 (x, ) = i=1 k=1 i=1 k=0 j=1 π 1,i,k x i k and the auxiliary generating functions ϕ 0,i () = π 0,i,k k, i 1, ϕ 1,i () = k=1 π 1,i,k k, i 1. k=0

5 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE 9 Multiplying (2), (3) by k and summing over k, these equations become ϕ 0,i () = pϕ 0,i+1 () + pa i ϕ 1,1 () pa i π 1,1,0, i 1, (4) ϕ 1,i () = ( p + p)ϕ 1,i+1 () + ps i ϕ 0 (1,)+ pa 0 + p s i ϕ 1,1 () + p s iϕ 0,1 () pa 0 s iπ 1,1,0 + ps i π 0,0, i 1. (5) By substituting equation (1) into equations (4), (5), we get ϕ 0,i () = pϕ 0,i+1 () + pa i ϕ 1,1 () pa i π 0,0, i 1, (6) ϕ 1,i () = ( p + p)ϕ 1,i+1 () + ps i ϕ 0 (1,)+ pa 0 + p s i ϕ 1,1 () + p s iϕ 0,1 () + p( a 0) s i π 0,0, i 1. (7) Then, multiplying (6), (7) by x i and summing over i leads to x p ϕ 0 (x, ) = p [ ] A(x) a 0 ϕ1,1 () pϕ 0,1 () p [ ] A(x) a 0 π0,0, (8) x [ ] x ( p + p) pa0 + p ϕ 1 (x, ) = S(x) ( p + p) ϕ 1,1 () x + p S(x)ϕ 0,1() + ps(x)ϕ 0 (1,)+ p( a 0) S(x)π 0,0. (9) Choosing x = 1in(8) yields pϕ 0 (1,)= p(1 a 0 )ϕ 1,1 () pϕ 0,1 () p(1 a 0 )π 0,0 and substituting the above equation into (9), weget [ ] x ( p + p) + pa0 (1 ) ϕ 1 (x, ) = S(x) ( p + p) ϕ 1,1 () x p(1 ) + S(x)ϕ 0,1 () pa 0(1 ) S(x)π 0,0. (10) Setting x = p in (8) and x = p + p in (10), we obtain p [ ] A( p) a 0 π0,0 = p [ ] A( p) a 0 ϕ1,1 () pϕ 0,1 (), [ ] pa 0 (1 ) + pa0 (1 ) S( p + p)π 0,0 = S( p + p) ( p + p) ϕ 1,1 () p(1 ) + S( p + p)ϕ 0,1 (). And from this system of equations, we find the generating functions ϕ 1,1 () and ϕ 0,1 (): ϕ 1,1 () = pa( p)(1 )S( p + p) pa( p)(1 )S( p + p) [( p + p) S( p + p)] π 0,0, (11)

6 10 ATENCIA AND MORENO ϕ 0,1 () = p[a( p) a 0 ][( p + p) S( p + p)] π 0,0 pa( p)(1 )S( p + p) [( p + p) S( p + p)] p. (12) Lemma 1. (1) pa( p)(1 )S( p + p) [( p + p) S( p + p)] > 0 holds for 0 <1if and only if ρ<p+ pa( p). (2) The following limits exist if and only if ρ<p+ pa( p): lim 1 lim 1 1 pa( p)(1 )S( p + p) [( p + p) S( p + p)] = 1 ( p + p) S( p + p) pa( p)(1 )S( p + p) [( p + p) S( p + p)] = p + pa( p) ρ, ρ p p + pa( p) ρ. Using lemma 1, the auxiliary generating functions ϕ 1,1 () and ϕ 0,1 () are defined for [0, 1) and in = 1 are extended by continuity if and only if ρ<p+ pa( p). Substituting (11), (12) into (8), (10), we have the generating functions A(x) A( p) px[( p + p) S( p + p)]π 0,0 ϕ 0 (x, ) = x p pa( p)(1 )S( p + p) [( p + p) S( p + p)], S(x) S( p + p) pxa( p)(1 )( p + p)π 0,0 ϕ 1 (x, ) = x ( p + p) pa( p)(1 )S( p + p) [( p + p) S( p + p)]. Using the normaliation condition π 0,0 + ϕ 0 (1, 1) + ϕ 1 (1, 1) = 1, we can find the unknown constant π 0,0 : p + pa( p) ρ π 0,0 =. (13) A( p) Obviously from (13), asπ 0,0 > 0, we obtain that ρ<p+ pa( p) is a necessary condition for the ergodicity of the Markov chain. Let us observe that this condition can be written as p(β 1 1) < pa( p) where the first member is the expected number of external customers who arrive per service interval and the second one represents the expected number of repeated customers who enter service at the epoch at which a service starts. Therefore, this condition indicates that external customers must arrive per service interval more slowly than repeated customers can enter service at the epoch at which a service starts (on the average). Accordingly, if the condition ρ<p+ pa( p) is fulfilled, then the system is stable and consequently this condition is also sufficient. We summarie the above results in the following theorem. Theorem 1. The Markov chain {Y m,m N} is ergodic if and only if ρ<p+ pa( p).

7 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE 11 The generating functions of the stationary distribution of the chain are given by A(x) A( p) px[( p + p) S( p + p)]π 0,0 ϕ 0 (x, ) = x p pa( p)(1 )S( p + p) [( p + p) S( p + p)], S(x) S( p + p) pxa( p)(1 )( p + p)π 0,0 ϕ 1 (x, ) = x ( p + p) pa( p)(1 )S( p + p) [( p + p) S( p + p)], where π 0,0 = p + pa( p) ρ. A( p) Corollary 1. (1) The marginal generating function of the number of customers in the retrial group when the server is idle is given by A( p)( p + p)[s( p + p) ]π 0,0 π 0,0 + ϕ 0 (1,)= pa( p)(1 )S( p + p) [( p + p) S( p + p)]. (2) The marginal generating function of the number of customers in the retrial group when the server is busy is given by A( p)( p + p)[1 S( p + p)]π 0,0 ϕ 1 (1,)= pa( p)(1 )S( p + p) [( p + p) S( p + p)]. (3) The probability generating function of the number of customers in the retrial group (i.e., of the variable N)isgivenby ()= π 0,0 + ϕ 0 (1,)+ ϕ 1 (1,) A( p)(1 )( p + p)π 0,0 = pa( p)(1 )S( p + p) [( p + p) S( p + p)]. (4) The probability generating function of the number of customers in the system (i.e., of the variable L)isgivenby () = π 0,0 + ϕ 0 (1,)+ ϕ 1 (1,) A( p)(1 )( p + p)s( p + p)π 0,0 = pa( p)(1 )S( p + p) [( p + p) S( p + p)]. In the next corollary we present some performance measures for the system at the stationary regime. Corollary 2. (1) The system is free with probability π 0,0 = p + pa( p) ρ. A( p)

8 12 ATENCIA AND MORENO (2) The system is occupied with probability ϕ 0 (1, 1) + ϕ 1 (1, 1) = (3) The server is idle with probability ρ p[1 A( p)]. A( p) (4) The server is busy with probability π 0,0 + ϕ 0 (1, 1) = 1 ρ. ϕ 1 (1, 1) = ρ. (5) The mean number of customers in the retrial group is E[N] = (1) = 2 p(ρ p)[1 A( p)]+p2 β 2. 2[p + pa( p) ρ] (6) The mean number of customers in the system is E[L] = (1) = ρ + 2 p(ρ p)[1 A( p)]+p2 β 2. 2[p + pa( p) ρ] (7) The mean time a customer spends in the system (including the service time) is given by W = E[L] p = β p(β 1 1)[1 A( p)]+pβ 2. 2[p + pa( p) ρ] The proof of both corollaries is trivial and thus omitted. Remark 1. The stationary distribution of the server state π 0,0 + ϕ 0 (1, 1) = 1 ρ, ϕ 1 (1, 1) = ρ, depends on the service time distribution only through its mean β 1 and does not depend on the interretrial time distribution. Remark 2. We observe a relation between the following generating functions () = ()S( p + p), and as a consequence we find the formula n ( ) n (n (1) = p m β m (n m (1), n 1, m m=0 where (n (1) and (n (1) are the nth factorial moments for the distribution of the random variables L and N, respectively.

9 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE 13 Remark 3 (Special case). When a 0 = 1, () reduces to (1 ρ)(1 )S( p + p) () =, S( p + p) which is the probability generating function for the number of customers (including the one in service, if any) in the standard Geo/G/1/ queueing system (see [10]). This result is not surprising because when a 0 = 1, the customer at the head of the orbit immediately commences his service whenever the server is idle. This system is equivalent to the Geo/G/1/ queue with random service order whose queue sie distribution is the same as that of the standard Geo/G/1/ queue since it is not affected by the service discipline adopted. 4. Stochastic decomposition This section deals with the analysis of the stochastic decomposition property of the system sie distribution. The first result on stochastic decomposition was given by Fuhrmann and Cooper [8]. Later, this property has been discussed by several authors. Generally speaking, the stochastic decomposition relates one performance characteristic for the system with vacations to the corresponding one for the same model without vacations. The stochastic decomposition property states that the system sie distribution decomposes into two random variables, one of which corresponds to the system sie of the queueing model without vacations. The second random variable is the system sie given that the server is on vacation. We now note that the probability generating function of the number of customers in the system can be expressed as () = (1 ρ)(1 )S( p + p) S( p + p) π 0,0 + ϕ 0 (1,) π 0,0 + ϕ 0 (1, 1), where the first fraction is the probability generating function of the number of customers in the Geo/G/1/ queueing system and the second fraction is the probability generating function of the number of customers in the orbit given that the server is idle. In fact, this is the stochastic decomposition law for our queueing system; i.e., the total number of customers in our queueing system can be represented by the sum of two independent random variables: one is the total number of customers in the corresponding regular system and the other is the number of customers in the orbit given that the server is idle. This result can be summaried in the following theorem. Theorem 2. The total number of customers in the system under study (L) can be represented as the sum of two independent random variables, one of which is the total number of customers in the Geo/G/1/ queueing system (L 0 ) and the other is the number of repeated customers given that the server is free (M). That is, L = L 0 + M.

10 14 ATENCIA AND MORENO It is not surprising that the system under study satisfies this property, since our system can be considered as a standard queue with server vacation. In this vacation model, the server begins vacation when a service concludes and there is neither arrival nor retrial. The duration of the vacation depends on the arrival process and the interretrial times. Vacation finish whenever an external customer arrives or the server chooses the customer at the head of the orbit. Under these considerations, the stochastic decomposition property observed previously for our system is consistent with the one reported by Fuhrmann and Cooper [8] for M/G/1 vacation models. As an application of the stochastic decomposition property we now study a measure of the proximity between the steady-state distributions for the standard Geo/G/1/ queueing system and our queueing system. The importance of the following bounds is to provide upper and lower estimates for the distance between both distributions. Theorem 3. The following inequalities hold (ρ p)[1 A( p)] 2 P [L = j] P [L0 = j] (ρ p)[1 A( p)] 2. A( p) (1 ρ)a( p) j=0 The proof of the preceding theorem follows the steps given in the paper [3] and accordingly it is omitted. Finally, let us observe that the distance j=0 P [L = j] P [L 0 = j] between the distributions of the random variables L and L 0 decreases as A( p) approaches Calculation of steady-state probabilities This section is devoted to develop some recursive formulae for calculating the more characteristic stationary distributions associated with our system. Theorem 4. The steady-state distribution of the orbit sie is given by the following recursive formulae p + pa( p) ρ ψ 0 = P [N = 0]=, (14) A( p)s( p) where ψ k = P [N = k]= k 1 n=0 [b k n pa( p)c k n ]ψ n, k 1, (15) A( p)s( p) ( ) i 1 b n = B i p i n p n, n 1, n 1 i=n ( ) i c n = s i+1 p i n p n, n 1, n i=n and B i = j=i+1 s j is the probability that a service lasts more than i slots.

11 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE 15 Proof. Let () be the function S( p + p) () = pa( p) p + p [ 1 1 ] S( p + p) = p + p ω n n. In order to obtain the sequence {ω n } n=0 we will use the properties of the generating functions and Newton s binomial (see [18, theorem 3]) getting [ ] () = A( p)s( p) bn pa( p)c n n. n=1 Since () () = p + pa( p) ρ, comparing the coefficients of k on both sides of this equation leads to k ψ 0 ω 0 = p + pa( p) ρ, ψ n ω k n = 0, k 1. But ω 0 = A( p)s( p) and ω n = pa( p)c n b n for n 1, and as consequence we find equations (14), (15). Theorem 5. The steady-state distribution of the system sie is given by the following recursive formulae p + pa( p) ρ φ 0 = P [L = 0]=, (16) A( p) k 1 n=0 φ k = P [L = k]= [b k n pa( p)c k n ]φ n +[p + pa( p) ρ]d k (17) A( p)s( p) for k 1, where ( ) i d n = s i p i n p n, n 1. n Proof. i=n Let us observe the equation n=0 () () = [ p + pa( p) ρ ] S( p + p), (18) where () and its expression in power series are given in the proof of the previous theorem. From the relation S( p + p) = S( p) + d n n and after comparing the coefficients of k on both sides in equation (18), we get n=1 φ 0 ω 0 = [ p + pa( p) ρ ] S( p), k φ n ω k n = [ p + pa( p) ρ ] d k, k 1. n=0 n=0

12 16 ATENCIA AND MORENO we obtain equa- Then, taking into account the expression of the sequence {ω n } n=0 tions (16), (17). In what continues, we will denote by π 0,,k = i=1 π 0,i,k the stationary probability that there are k 1 repeated customers and the server is idle, and π 1,,k = i=1 π 1,i,k the stationary probability that there are k 0 repeated customers and the server is busy. Theorem 6. The steady-state distribution of the orbit sie when the server is idle is given by the following recursive formulae where p + pa( p) ρ π 0,0 =, A( p) π 0,,k = [b k pa( p)c k ]π 0,0 + (1 δ 1k ) k 1 n=1 [b k n pa( p)c k n ]π 0,,n A( p)s( p) p + pa( p) ρ e k, k 1, A( p)s( p) e n = i=n ( ) i B i p i n p n+1, n 1. n Proof. Firstly, we observe the equation [ π0,0 + ϕ 0 (1,) ] () = [ p + pa( p) ρ ] S( p + p), (19) 1 where the function () and its development in power series are given in the proof of the theorem 4. From the equality S( p + p) 1 = S( p) e n n and after equaliing the coefficients of k on both sides of equation (19), we have π 0,0 ω k + n=1 π 0,0 ω 0 = [ p + pa( p) ρ ] S( p), k π 0,,k ω k n = [ p + pa( p) ρ ] e k, k 1. n=1 Finally, considering the expression of the sequence {ω n } n=0, the proof of this theorem is completed. Theorem 7. The steady-state distribution of the orbit sie when the server is busy is given by the following recursive formulae

13 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE 17 p + pa( p) ρ π 1,,0 = A( p) π 1,,k = k 1 1 S( p), S( p) n=0 [b k n pa( p)c k n ]π 1,,n +[p + pa( p) ρ]e k, k 1. A( p)s( p) Proof. We now observe the following equation ϕ 1 (1, ) () = [ p + pa( p) ρ ] 1 S( p + p), (20) 1 where the definition of () and its expression in power series are given in the proof of the theorem 4. Since 1 S( p + p) 1 = 1 S( p) + e n n, n=1 equalling the coefficients of k on both sides of equation (20) yields π 1,,0 ω 0 = [ p + pa( p) ρ ][ 1 S( p) ], k π 1,,n ω k n = [ p + pa( p) ρ ] e k, k 1. n=0 To conclude this proof, it will be enough to keep in mind the expression of the sequence {ω n } n=0. Remark 4. If the service times are geometrically distributed with generating function S(x) = (1 s)x/(1 sx), then the coefficients b n, c n, d n and e n have the following closed-form expressions for n 1. b n = pn s n (1 ps), c n n = pn s n (1 s) (1 ps), n+1 d n = pn s n 1 (1 s) (1 ps) n+1, e n = pn+1 s n (1 ps) n+1 6. Relation with the continuous-time system This section concerns the analysis of the relation between the continuous-time system and the discrete-time system. We will show that the continuous-time M/G/1 retrial queue with general retrial times can be approximated by the corresponding discrete-

14 18 ATENCIA AND MORENO time system; for this purpose, time is slotted into small intervals of equal length, so the approximation approaches the exact value when the length of the interval tends to ero. We consider the continuous-time M/G/1 retrial queue with general retrial times (see [9] for details) where customers arrive according to a Poisson process with rate λ. Upon arrival, the customer who finds the server busy leaves the service area and joins the retrial group in accordance with an FCFS discipline (that is, only the customer at the head of the orbit is allowed for access to the server). Successive interretrial times of any customer are governed by an arbitrary probability distribution function Ɣ(x) with corresponding Laplace Stieltjes transform γ(s). Customer service times are identically and independently distributed random variables with a common distribution function B(x), Laplace Stieltjes transform β(s) and a finite mean µ 1. Interarrival times, retrial times and service times are assumed to be mutually independent. If we suppose that time is divided into intervals of equal length, the continuoustime system can be approximated by a discrete-time system for which p = λ, a i = (i+1) i dɣ(x), i 0 and s i = i (i 1) db(x), i 1, where must be chosen sufficiently small so that p is probability. Our immediate objective is to prove that lim 0 () is the probability generating function of the number of customers in the M/G/1 retrial queueing system with general retrial times (obtained by Góme-Corral [9]). Firstly, it is not difficult to prove the following equalities using the definition of Lebesgue integration: lim 0 lim ρ = 0 λµ 1, A( p) = γ(λ), lim S( p + p) = β( λ(1 ) ). 0 The proof of these relationships is omitted here since the technique used can be found in [18, theorem 5]. Taking into account the above results, we get the next relation lim () = lim 0 0 (1 )( p + p)s( p + p)[p + pa( p) ρ] pa( p)(1 )S( p + p) [( p + p) S( p + p)] (1 )(1 λ(1 ) )S( p + p)[λ + (1 λ )A( p) ρ] = lim 0 (1 λ )A( p)(1 )S( p + p) [1 λ(1 ) S( p + p)] [ = γ(λ) λ ] (1 )β(λ(1 )) µ γ(λ)(1 )β(λ(1 )) [1 β(λ(1 ))], which coincides with the probability generating function of the number of customers in the M/G/1 retrial queue with general retrial times (see [9, equation (16)]).

15 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE Numerical results In this section, we present some numerical results to illustrate the effect of varying parameters on the main performance measures of our system. In figures 1(a) and (b), we consider that the service time distribution is geometrical with mean β 1 = 2 and the retrial times are governed by a geometrical distribution with generating function A(x) = (1 r)/(1 rx). In figure 1(a) the probability that the system is busy is plotted versus the retrial rate r. We have presented three curves which correspond to p = 0.2, 0.3, 0.4, respectively. As we expect, the utiliation factor increases with increasing retrial rate r and increasing p. The same discussion holds for figure 1(b), which illustrates the behaviour of E[L] as function of r. Moreover, we observe that, as r approaches the ergodicity condition, the mean system sie tends to infinite (due to the system becomes unstable) and, as a consequence, the probability that the system is occupied converges to 1. In figures 1(c) and (d), we assume that the parameter p is equal to 0.2, the service times follow a geometrical distribution with mean β 1 = 2 and the retrial times have a binomial distribution with generating function A(x) = (rx + 1 r) n. (a) (b) (c) (d) Figure 1. (a) The utiliation factor versus r. (b) The mean system sie versus r. (c) The utiliation factor versus r. (d) The mean system sie versus r.

16 20 ATENCIA AND MORENO The influence of the parameter r on the utiliation factor is shown in figure 1(c). The highest curve in figure 1(c) corresponds to the highest value of n. As intuition tells us, the probability that the system is busy increases with increasing values of n. From this graphic, we see that the utiliation factor increases with increasing values of r,which also agrees with the intuitive expectations. Substantially, the same effects are shown in figure 1(d), which represents the behaviour of E[L] as function of r. We would like to remark that both the utiliation factor and the mean system sie are consistent (when r 0) with the characteristics of the standard Geo/G/1/ queue given by ρ = 0.4 ande[l] = , respectively, that is, we have the limiting results [ lim ϕ0 (1, 1) + ϕ 1 (1, 1) ] = ρ and lim E[L] =ρ + p2 β 2 r 0 r 0 2(1 ρ). Besides, we note that the utiliation factor and the mean system sie in the classi- Table 1 The steady-state distribution of the orbit sie. r = 0 r = 0.3 r = 0.6 r = 0.9 ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ Table 2 The steady-state distribution of the system sie. r = 0 r = 0.3 r = 0.6 r = 0.9 φ φ φ φ φ φ φ φ φ φ φ

17 DISCRETE-TIME Geo/G/1 RETRIAL QUEUE 21 cal Geo/G/1/ queue are a lower bound for the corresponding characteristics in our queueing system. An important feature of this work is in the recursion scheme provided by theorems 4 and 5. The formulae (14) (17) have been implemented in Matlab. Tables 1 and 2 summarie our numerical experiments for p = 0.2, service times geometrically distributed with mean β 1 = 2 and retrial times governed by a geometrical distribution with generating function A(x) = (1 r)/(1 rx). Acknowledgement This research is supported by the DGINV through the project BFM References [1] J.R. Artalejo, A classified bibliography of research on retrial queues: Progress in , Top 7(2) (1999) [2] J.R. Artalejo, Accessible bibliography on retrial queues, Math. Comput. Modelling 30 (1999) 1 6. [3] J.R. Artalejo and G.I. Falin, Stochastic decomposition for retrial queues, Top 2 (1994) [4] I. Atencia and P. Moreno, Discrete-time Geo [X] /G H /1 retrial queue with Bernoulli feedback, To appear in Comput. Math. Appl. [5] H. Bruneel and B.G. Kim, Discrete-Time Models for Communication Systems Including ATM (Kluwer Academic, Boston, 1993). [6] B.D. Choi and J.W. Kim, Discrete-time Geo 1, Geo 2 /G/1 retrial queueing system with two types of calls, Comput. Math. Appl. 33(10) (1997) [7] G.I. Falin and J.G.C. Templeton, Retrial Queues (Chapman & Hall, London, 1997). [8] S.W. Fuhrmann and R.B. Cooper, Stochastic decompositions in the M/G/1 queue with generalied vacations, Oper. Res. 33(5) (1985) [9] A. Góme-Corral, Stochastic analysis of a single server retrial queue with general retrial times, Naval Res. Logistics 46 (1999) [10] J.J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete-Time Models: Techniques and Applications (Academic Press, New York, 1983). [11] H. Li and T. Yang, Geo/G/1 discrete time retrial queue with Bernoulli schedule, European J. Oper. Res. 111(3) (1998) [12] H. Li and T. Yang, Steady-state queue sie distribution of discrete-time PH/Geo/1 retrial queues, Math. Comput. Modelling 30 (1999) [13] T. Meisling, Discrete time queueing theory, Oper. Res. 6 (1958) [14] M.F. Neuts and M.F. Ramalhoto, A service model in which the server is required to search for customers, J. Appl. Probab. 21 (1984) [15] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Discrete-Time Systems, Vol. 3 (North-Holland, Amsterdam, 1993). [16] M. Takahashi, H. Osawa and T. Fujisawa, Geo [X] /G/1 retrial queue with non-preemptive priority, Asia-Pacific J. Oper. Res. 16(2) (1999) [17] M.E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues (IEEE Computer Soc. Press, Los Alamitos, CA, 1994). [18] T. Yang and H. Li, On the steady-state queue sie distribution of the discrete-time Geo/G/1 queue with repeated customers, Queueing Systems 21 (1995)

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