Sojourn time asymptotics in the M/G/1 processor sharing queue

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

M/G/1/MLPS compared with M/G/1/PS within service time distribution class IMRL

Multilevel processor-sharing (MLPS) disciplines were originally introduced by Kleinrock (in computer applications 1976) but they were forgotten for years. However, due to an application related to the service differentiation between short and long TCP flows in the Internet, they have recently gained new interest. In this paper we show that, if the service time distribution belongs to class IMRL, the mean delay in the M/G/1 queue is reduced when replacing the PS discipline with any MLPS discipline for which the internal disciplines belong to {FB, PS}. This is a generalization of our earlier result where we restricted ourselves to the service time distribution class DHR, which is a subset of class IMRL.     

The MMCPP/GE/c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.     

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.     

On the M/G/2 queeing model

The M/G/2 queueing model with service time distribution a mixture of m negative exponential distributions is analysed. The starting point is the functional relation for the Laplace–Stieltjes transform of the stationary joint distribution of the workloads of the two servers. By means of Wiener–Hopf decompositions the solution is constructed and reduced to the solution of m linear equations of which the coefficients depend on the zeros of a polynome. Once this set of equations has been solved the moments of the waiting time distribution can be easily obtained. The Laplace–Stieltjes transform of the stationary waiting time distribution has been derived, it is an intricate expression.     

Comparison conjectures about the M/G/s queue

We conjecture that the equilibrium waiting-time distribution in an M/G/s queue increases stochastically when the service-time distribution becomes more variable. We discuss evidence in support of this conjecture and others based partly on light-traffic and heavy-traffic limits. We also establish an insensitivity property for the case of many servers in light traffic.     

Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions

We consider a GI/G/1 queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like t ^−ν with 1 < ν ⩽ 2 , so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time W. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load a → 1, then W, multiplied by an appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load a → 1, then W, multiplied by another appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the negative exponential distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Derivation of the N-step interdeparture time distribution in GI/G/1 queueing systems

The departure process of a queueing system has been studied since the 1960s. Due to its inherent complexity, closed form solutions for the distribution of the departure process are nearly intractable. In this paper, we derive a closed form expression for the distribution of interdeparture time in a GI/G/1 queueing model. Without loss of generality, we consider an embedded Markov chain in a general K_M/G/1 queueing system, in which the interarrival time distribution is Coxian and service time distribution is general. Closed form solutions of the equilibrium distribution are derived for this model and the Laplace–Stieltjes transform (LST) of the distribution of interdeparture times is presented. An algorithmic computing procedure is given and numerical examples are provided to illustrate the results. With the analysis presented, we provide a novel analytic tool for studying the departure process in a general queueing model.     

对M/T-SPH/1排队平稳队长的分析

研究了M/T-SPH/l排队模型,利用拟生灭过程和算子几何解的方法给出了平稳队长分布的概率母函数.在此基础上,指出该分布不是一个离散PH分布,但在一定条件下却是一个几何尾部分布.     

Steady state analysis of an M/G/1 queue with two phase service and Bernoulli vacation schedule under multiple vacation policy

This paper examines the steady state behaviour of a batch arrival queue with two phases of heterogeneous service along and Bernoulli schedule vacation under multiple vacation policy, where after two successive phases service or first vacation the server may go for further vacations until it finds a new batch of customer in the system. We carry out an extensive stationary analysis of the system, including existence of stationary regime, queue size distribution of idle period process, embedded Markov chain steady state distribution of stationary queue size, busy period distribution along with some system characteristics.     

AnM/M/1 retrial queue with control policy and general retrial times

We consider anM/M/1 retrial queueing system in which the retrial time has a general distribution and only the customer at the head of the queue is allowed to retry for service. We find a necessary and sufficient condition for ergodicity and, when this is satisfied, the generating function of the distribution of the number of customers in the queue and the Laplace transform of the waiting time distribution under steady-state conditions. The results agree with known results for special cases.Supported by KOSEF 90-08-00-02.     

A Retrial BMAP/PH/N System

A multi-server retrial queueing model with Batch Markovian Arrival Process and phase-type service time distribution is analyzed. The continuous-time multi-dimensional Markov chain describing the behavior of the system is investigated by means of reducing it to the corresponding discrete-time multi-dimensional Markov chain. The latter belongs to the class of multi-dimensional quasi-Toeplitz Markov chains in the case of a constant retrial rate and to the class of multi-dimensional asymptotically quasi-Toeplitz Markov chains in the case of an infinitely increasing retrial rate. It allows to obtain the existence conditions for the stationary distribution and to elaborate the algorithms for calculating the stationary state probabilities.     

GI/M/1 Queue with Server Vacations

A queueing model with server vacations is studied in which it is assumed that the interarrival time has a general distribution, the service-time distribution is exponential and vacations are independently and indentically distributed with a general distribution. Using the embedded Markov chain technique, the equilibrium probability distributions of system size have been obtained at pre-arrival and at random epochs separately. Finally, the distribution of waiting time of a customer in the queue (excluding service) has been derived.     

M/C2/c/K queuing model and optimization for a geriatric care center

Queuing systems with finite buffers are reasonable models for many manufacturing, telecommunication, and healthcare systems. Although some approximations exist, the exact analysis of multi‐server and finite‐buffer queues with general service time distribution is unknown. However, the phase‐type assumption for service time is a frequently used approach. Because the Cox distribution, a kind of phase‐type distribution, provides a good representation of data with great variability, it has a vast area of application in modeling service times. The research focus is twofold. First, a theoretical structure of a multi‐server and finite‐buffer queuing system in which the service time is modeled by the two‐phase Cox distribution is studied. It is focused on finding an efficient solution to the stationary probabilities using the matrix‐geometric method. It is shown that the stationary probability vector can be obtained with the matrix‐geometric method by using level‐dependent rate matrices, and the mean queue length is computed. Second, an empirical analysis of the model is presented. The proposed methodology is applied in a case study concerning the geriatric patients. Some numerical calculations and optimizations are performed by using geriatric data. Copyright © 2015 John Wiley & Sons, Ltd.     

Equilibrium results for the M/G/k group-arrival loss system

In this paper we consider a specific M/G/k group-arrival loss system, under statistical equilibrium and two cases of acceptance policy. In the first case the system works under the partial acceptance policy. Explicit results are obtained for the corresponding stationary distribution, which extend previous relevant results. In the second case, where the system works under the all-or-nothing acceptance policy, a sufficient condition is given for the stationary distribution to have a closed-product form. In both cases customers depart individually, while the joint service time distribution of the accepted members of a group may depend both on its initial and the accepted size plus an additional condition.     

Computational analysis of single-server bulk-arrival queues: GIX/M/1

This paper deals with numerical computations for the bulk-arrival queueing modelGI ^X/M/1. First an algorithm is developed to find the roots inside the unit circle of the characteristic equation for this model. These roots are then used to calculate both the moments and the steady-state distribution of the number of customers in the system at a pre-arrival epoch. These results are used to compute the distribution of the same random variable at post-departure and random epochs. Unifying the method used by Easton [7], we have extended its application to the special cases where the interarrival time distribution is deterministic or uniform, and to cases whereX has a given arbitrary distribution. We also improved on the various root-finding methods used by several previous authors so that high values of the parameters, in particular large batch sizes, can be investigated as well.     

TheM/GI/1 Bernoulli feedback queue with vacations

Feedback may be introduced as a mechanism for scheduling customer service (for example in systems in which customers bring work that is divided into a random number of stages). A model is developed that characterizes the queue length distribution as seen following vacations and service stage completions. We demonstrate the relationship that exists between these distributions. The ergodic waiting time distribution is formulated in such a way as to reveal the effects of server vacations when feedback is introduced.This work was supported in part by NSF Grant No. DDM-8913658.     

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.