Duality relations for certain single server queues

In this paper, relations are given between the joint distribution of several variables in a GI/G/1 queue and the joint distribution of variables associated with the busy cycle in the dual queue, that is in the queue which results from the original when the interarrival times and the service times are interchanged. It is assumed that the primal queue has the preemptive-resume last-come-first-served queue discipline while the dual queue may have any queue discipline which is work conserving. These relations generalize a result given recently for M/G/1 and GI/M/1 queues.     

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.     

Conditions for finite moments of waiting times inG/G/1 queues

We find conditions for E(W ) to be finite whereW is the stationary waiting time random variable in a stableG/G/1 queue with dependent service and inter-arrival times.Supported in part by KBN under grant 640/2/9, and at the Center for Stochastic Processes, Department of Statistics at the University of North Carolina Chapel Hill by the Air Force Office of Scientific Research Grant No. 91-0030 and the Army Research Office Grant No. DAAL09-92-G-0008.     

Settling time bounds forM/G/1 queues

This paper addresses the question of how long it takes for anM/G/1 queue, starting empty, to approach steady state. A coupling technique is used to derive bounds on the variation distance between the distribution of number in the system at timet and its stationary distribution. The bounds are valid for allt. This research was supported in part by a grant from the AT&T Foundation and NSF grant DCR-8351757.     

A sample path relation for the sojourn times in G/G/1−PS systems and its applications

For the single server system under processor sharing (PS) a sample path result for the sojourn times in a busy period is proved, which yields a sample path relation between the sojourn times under PS and FCFS discipline. This relation provides a corresponding one between the mean stationary sojourn times in G/G/1 under PS and FCFS. In particular, the mean stationary sojourn time in G/D/1 under PS is given in terms of the mean stationary sojourn time under FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of these results suggest an approximation of the mean stationary sojourn time in G/GI/1 under PS in terms of the mean stationary sojourn time under FCFS. Mathematics Subject Classification (MSC 2000) 60K25· 68M20· 60G17· 60G10 This work was supported by a grant from the Siemens AG.     

Clearing Models for M/G/1 Queues

We consider M/G/1-type queueing systems with disasters, occurring at certain random times and causing an instantaneous removal of the entire residual workload from the system. After such a clearing, the system is assumed to be ready to start working again immediately. We consider clearings at deterministic equidistant times, at random times and at crossings of some prespecified level, and derive the stationary distribution of the workload process for these clearing times and some of their combinations.     

The Virtual Waiting Time of the M/G/1 Queue with Impatient Customers

The M/G/1 queue with impatient customers is studied. The complete formula of the limiting distribution of the virtual waiting time is derived explicitly. The expected busy period of the queue is also obtained by using a martingale argument.     

Analyticity of single-server queues in light traffic

Recently, several methods have been proposed to approximate performance measures of queueing systems based on their light traffic derivatives, e.g., the MacLaurin expansion, the Padé approximation, and interpolation with heavy traffic limits. The key condition required in all these approximations is that the performance measures be analytic when the arrival rates equal to zero. In this paper, we study theGI/G/1 queue. We show that if the c.d.f. of the interarrival time can be expressed as a MacLaurin series over [0, ), then the mean steady-state system time of a job is indeed analytic when the arrival rate to the queue equals to zero. This condition is satisfied by phase-type distributions but not c.d.f.'s without support [0, ), such as uniform and shifted exponential distributions. In fact, we show through two examples that the analyticity does not hold for most commonly used distribution functions which do not satisfy this condition.     

New bounds for expected delay in FIFO M/G/c queues

Most bounds for expected delay, E[D], in GI/GI/c queues are modifications of bounds for the GI/GI/1 case. In this paper we exploit a new delay recursion for the GI/GI/c queue to produce bounds of a different sort when the traffic intensity p = λ/μ = E[S]/E[T] is less than the integer portion of the number of servers divided by two. (S AND T denote generic service and interarrival times, respectively.) We derive two different families of new bounds for expected delay, both in terms of moments of S AND T. Our first bound is applicable when E[S₂] < ∞. Our second bound for the first time does not require finite variance of S; it only involves terms of the form E[S^β], where 1 < β < 2. We conclude by comparing our bounds to the best known bound of this type, as well as values obtained from simulation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximations for waiting time in GI/G/1 systems

The Sokolov procedure is described and used to obtain an explicit and easily applied approximation for the waiting time distribution in the FIFO GI/G/1 queue.     

Delay moments for FIFO GI/GI/s queues

For stable FIFO GI/GI/s queues, s ≥ 2, we show that finite (k+1)st moment of service time, S, is not in general necessary for finite kth moment of steady-state customer delay, D, thus weakening some classical conditions of Kiefer and Wolfowitz (1956). Further, we demonstrate that the conditions required for E[D ^k]<∞ are closely related to the magnitude of traffic intensity ρ (defined to be the ratio of the expected service time to the expected interarrival time). In particular, if ρ is less than the integer part of s/2, then E[D] < ∞ if E[S^3/2]<∞, and E[D^k]<∞ if E[S^k]<∞, ^k≥ 2. On the other hand, if s-1 < ρ < s, then E[D^k]<∞ if and only if E[S^k+1]<∞, k ≥ 1. Our method of proof involves three key elements: a novel recursion for delay which reduces the problem to that of a reflected random walk with dependent increments, a new theorem for proving the existence of finite moments of the steady-state distribution of reflected random walks with stationary increments, and use of the classic Kiefer and Wolfowitz conditions. This revised version was published online in June 2006 with corrections to the Cover Date.     

The M/G/1 Queue with Finite Workload Capacity

We consider the M/G/1 queueing system in which customers whose admission to the system would increase the workload beyond a prespecified finite capacity limit are not accepted. Various results on the distribution of the workload are derived; in particular, we give explicit formulas for its stationary distribution for M/M/1 and in the general case, under the preemptive LIFO discipline, for the joint stationary distribution of the number of customers in the system and their residual service times. Furthermore, the Laplace transform of the length of a busy period is determined. Finally, for M/D/1 the busy period distribution is derived in closed form.     

Conditional and unconditional distributions forM/G/1 type queues with server vacations

M/G/1 queues with server vacations have been studied extensively over the last two decades. Recent surveys by Boxma [3], Doshi [5] and Teghem [14] provide extensive summary of literature on this subject. More recently, Shanthikumar [11] has generalized some of the results toM/G/1 type queues in which the arrival pattern during the vacations may be different from that during the time the server is actually working. In particular, the queue length at the departure epoch is shown to decompose into two independent random variables, one of which is the queue length at the departure epoch (arrival epoch, steady state) in the correspondingM/G/1 queue without vacations. Such generalizations are important in the analysis of situations involving reneging, balking and finite buffer cyclic server queues. In this paper we consider models similar to the one in Shanthikumar [11] but use the work in the system as the starting point of our investigation. We analyze the busy and idle periods separately and get conditional distributions of work in the system, queue length and, in some cases, waiting time. We then remove the conditioning to get the steady state distributions. Besides deriving the new steady state results and conditional waiting time and queue length distributions, we demonstrate that the results of Boxma and Groenendijk [2] follow as special cases. We also provide an alternative approach to deriving Shanthikumar's [11] results for queue length at departure epochs.     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

Second-Order Tail Behavior of the Busy Period Distribution of Certain GI/G/1 Queues

We consider the stable GI/G/1 queue in which the service time distribution has a dominated-varying tail. Under simple assumptions, we obtain the first- and second-order tail behavior of the busy period distribution in this queue.     

Consistency of infinitesimal perturbation analysis estimators with rates

We study a class of infinitesimal perturbation analysis (IPA) algorithms for queueing systems with load-dependent service and/or arrival rates. Such IPA algorithms were originally motivated by applications to large queueing systems in conjunction with aggregation algorithms. We prove strong consistency of these estimators through a type of birth and death queue. This work was supported in part by the NSF under Grants Nos. ECS85-15449 and CDR-8803012, by ONR under Contracts Nos. N00014-89-J-0075 and N00014-90-K-1093, and by the US Army under Contract No. DAAL-03-83-K-0171. This paper was written while the author was with the Division of Applied Sciences at Harvard University.     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

On the longest service time in a busy period of the M⧸G⧸1 queue

This paper considers the supremum m of the service times of the customers served in a busy period of the M?G?1 queueing system. An implicit expression for the distribution m(w) of m is derived. This expression leads to some bounds for m(w), while it can also be used to obtain numerical results. The tail behaviour of m(w) is investigated, too. The results are particularly useful in the analysis of a class of tandem queueing systems.     

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.     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.