The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM _t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.     

Approximations for multi-server queues: System interpolations

This paper provides a unifying method of generating and/or evaluating approximations for the principal congestion measures in aGI/G/s queueing system. The main focus is on the mean waiting time, but approximations are also developed for the queue-length distribution, the waiting-time distribution and the delay probability for the Poisson arrival case. The approximations have closed forms that combine analytical solutions of simpler systems, and hence they are referred to as system-interpolation approximations or, simply, system interpolations. The method in this paper is consistent with and generalizes system interpolations previously presented for the mean waiting time in theGI/G/s queue.     

Approximating the Mean Waiting Time in the <Emphasis Type="Italic">GI/G/s</Emphasis> Queue

We provide two distribution-dependent approximations for the mean waiting time in a GI/G/s queue. Both approximations are weighted combinations of the exact mean waiting times for the GI/M/s and M/D/s queues each of which has the same mean service time and traffic intensity as in the approximating GI/G/s queue. The weights in the approximations are expressed by the service-time c.d.f. and the first two moments of interarrival and service times. To examine the performance of our approximations, they are numerically compared with exact solutions and previous two-moment approximations for various cases. Extensive numerical comparisons indicate that the relative percentage errors of the approximations are of the order of 5% in moderate traffic and 1% in heavy traffic, except for extreme cases.     

AnM/M/c queue with impatient customers

This paper considers theM/M/c queue in which a customer leaves when its service has not begun within a fixed interval after its arrival. The loss probability can be expressed in a simple formula involving the waiting time probabilities in the standardM/M/c queue. The purpose of this paper is to give a probabilistic derivation of this formula and to outline a possible use of this general formula in theM/M/c retrial queue with impatient customers. This research was supported by the INTAS 96-0828 research project and was presented at the First International Workshop on Retrial Queues, Universidad Complutense de Madrid, Madrid, September 22–24, 1998.     

Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

Two variants of an M/G/1 queue with negative customers lead to the study of a random walkX _n+1=[X _n + _n ]⁺ where the integer-valued _n are not bounded from below or from above, and are distributed differently in the interior of the state-space and on the boundary. Their generating functions are assumed to be rational. We give a simple closed-form formula for , corresponding to a representation of the data which is suitable for the queueing model. Alternative representations and derivations are discussed. With this formula, we calculate the queue length generating function of an M/G/1 queue with negative customers, in which the negative customers can remove ordinary customers only at the end of a service. If the service is exponential, the arbitrarytime queue length distribution is a mixture of two geometrical distributions.Supported by the European grant BRA-QMIPS of CEC DG XIII.     

A consistent diffusion approximation for finite-capacity multiserver queues

A diffusion approximation is developed for general multiserver queues with finite waiting spaces, which are typical models of manufacturing systems as well as computer and communication systems. The model is the standard GI/G/s/s + r queue with s identical servers in parallel, r extra waiting spaces, and the first-come, first-served discipline. The main focus is on the steady-state distribution of the number of customers in the system. The process of the number of customers is approximated by a time-homogeneous diffusion process on a closed interval in the nonnegative real line. A conservation law plus some heuristics standing on solid theoretical ground generate approximation formulas for the steady-state distribution and other congestion measures. These formulas are consistent with the exact results for the M/G/s/s and M/M/s/s + r queues. The accuracy of approximations for principal congestion measures are numerically examined for some particular cases.     

On some relations between stationary time and customer state probabilities for qneueing systems G/GI/s/r

By means of a general formula for stochastic processes with imbedded marked point processes (PMP) some necessary and sufficient condition is given for the validity of a relationship, which is well-known in the case of exponentially distributed service times, between stationary time and customer state probabilities for loss systems G/GI/s/O (Theorem 3). A result of Miyazawa for the GI/GI/l/∞ queue is generalized to the case of non-recurrent interarrival times (Theorem 4)-. Furthermore, bounds are derived for the mean increment of the waiting time process at arrival epochs and for the mean actual waiting time in multi-server queues.     

Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline

We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: we study the long-run state behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both actual and virtual delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural explanations for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Bene's formula for the delay distribution in M/G/l. We also discuss how to extend our results to settings more general than GI/G/1.     

On the GI/M/∞ queue with batch arrivals of constant size

In this note, the GI/M/ queue with batch arrivals of constant sizek is investigated. It is shown that the stationary probabilities that an arriving batch findsi customers in the system can be computed in terms of the corresponding binomial moments (Jordan's formula), which are determined by a recursive relation. This generalizes well-known results by Takács [12] for GI/M/. Furthermore, relations between batch arrival- and time-stationary probabilities are given.     

Equilibrium properties of the M/G/1 queue

Summary Various aspects of the equilibrium M/G/1 queue at large values are studied subject to a condition on the service time distribution closely related to the tail to decrease exponentially fast. A simple case considered is the supplementary variables (age and residual life of the current service period), the distribution of which conditioned upon queue length n is shown to have a limit as n. Similar results hold when conditioning upon large virtual waiting times. More generally, a number of results are given which describe the input and output streams prior to large values e.g. in the sense of weak convergence of the associated point processes and incremental processes. Typically, the behaviour is shown to be that of a different transient M/G/1 queueing model with a certain stochastically larger service time distribution and a larger arrival intensity. The basis of the asymptotic results is a geometrical approximation for the tail of the equilibrium queue length distribution, pointed out here for the GI/G/1 queue as well.     

A unified algorithm for computing the stationary queue length distributions inM(k)/G/1/N and GI/M(k)/1/N queues

An iterative algorithm is developed for computing numerically the stationary queue length distributions in M/G/1/N queues with arbitrary state-dependent arrivals, or simply M(k)/G/1/N queues. The only input requirement is the Laplace-Stieltjes transform of the service time distribution.In addition, the algorithm can also be used to obtain the stationary queue length distributions in GI/M/1/N queues with state-dependent services, orGI/M(k)/1/N, after establishing a relationship between the stationary queue length distributions inGI/M(k)/1/N and M(k)/G/1/N+1 queues.Finally, we elaborate on some of the well studied special cases, such asM/G/1/N queues,M/G/1/N queues with distinct arrival rates (which includes the machine interference problems), andGI/M/C/N queues. The discussions lead to a simplified algorithm for each of the three cases.     

A tandem Jackson network with feedback to the first node

AnN-node tandem queueing network with Bernoulli feedback to the end of the queue of thefirst node is considered. We first revisit the single-nodeM/G/1 queue with Bernoulli feedback, and derive a formula forEL(n), the expected queue length seen by a customer at his nth feedback. We show that, asn becomes large,EL(n) tends to /(l ), being the effective traffic intensity. We then treat the entire queueing network and calculate the mean value ofS, the total sojourn time of a customer in theN-node system. Based on these results we study the problem ofoptimally ordering the nodes so as to minimize ES. We show that this is a special case of a general sequencing problem and derive sufficient conditions for an optimal ordering. A few extensions of the serial queueing model are also analyzed. We conclude with an appendix in which we derive an explicit formula for the correlation coefficient between the number of customers seen by an arbitrary arrival to anM/G/1 queue, and the number of customers he leaves behind him upon departure. For theM/M/1 queue this coefficient simply equals the traffic intensity .     

Fluid queue driven by aGI/GI/1 queue

Consider a model consisting of two phases: the GI/GI/1 queue and a buffer which is fed by a fluid arriving from a single-server queue. The fluid output from the GI/GI/1 queue is of the on/off type with on- and off-periods distributed as successive busy and idle periods in the GI/GI/1 queue. The fluid pours out of the buffer at a constant rate. The steady-state performance of this model is studied. We derive the Laplace-Stieltjes transform of the stationary distribution function of the buffer content in the case of the M/GI/1 queue in the first phase. It is shown that this distribution depends on the form of the service-time distribution. Therefore, the replacement of an M/GI/1 queue by an M/M/1 queue is not correct, in general. Continuity estimates are derived in the cast where the buffer is fed from the GI/GI/1 queue. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow Russia, 1996, Part II.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

On stationary queue length distributions for G/M/s/r queues

Consider aG/M/s/r queue, where the sequence{A _n } _n=– of nonnegative interarrival times is stationary and ergodic, and the service timesS _n are i.i.d. exponentially distributed. (SinceA _n =0 is possible for somen, batch arrivals are included.) In caser < , a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr= (with=ES₀/EA₀<s) and Franken et al. [9], Borovkov [2] forr=0 ors=. Furthermore, we give a proof of the relation min(i, s)¯p(i)=p(i–1), 1ir + s, between the time- and arrival-stationary probabilities¯p(i) andp(i), respectively. This extends earlier results of Franken [7], Franken et al. [9].     

Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time

We consider anM/G/1 queue with FCFS queue discipline. We present asymptotic expansions for tail probabilities of the stationary waiting time when the service time distribution is longtailed and we discuss an extension of our methods to theM ^[x]/G/1 queue with batch arrivals.     

Approximations for the delay probability in the M/G/s queue

This paper develops approximations for the delay probability in an M/G/s queue. For M/G/s queues, it has been well known that the delay probability in the M/M/s queue, i.e., the Erlang delay formula, is usually a good approximation for other service-time distributions. By using an excellent approximation for the mean waiting time in the M/G/s queue, we provide more accurate approximations of the delay probability for small values of s. To test the quality of our approximations, we compare them with the exact value and the Erlang delay formula for some particular cases.     

Non-parametric Estimation for the M/G/∞ Queue

Given an M/G/ queue with input rate and service-time distribution G, we consider the problem of estimating and G from data on the queue-length process Q = (Q_t). Our motivation is to study departures of G from exponentiality, following recent work of Bingham and Dunham (1997, Ann. Inst. Statist. Math., 49, 667–679).     

Operating characteristics of MX/G/1 queue with N-policy

We consider aM ^X/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theM ^X/G/1 queueing system withoutN-policy and the other one has the probability generating function _j=0 ^N=1 _j z ^j/ _j=0 ^N=1 _j , in which _j is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.