Stationary Waiting Time Distribution Research Articles

A level-crossing approach to the solution of the shortest-queue problem

We develop a two-dimensional extension of the standard level-crossing approach and apply it to the shortest-queue problem. Here, the criterion for selecting a queue to join is based on a comparison of the accumulated workload in each of the queues. In particular, we obtain a simple closed-form expression for the stationary waiting-time distribution for the asymmetric two-server shortest-queue system. This result is then generalized to the many server case. Analytical forms for characteristics such as moments of the waiting time and queue length are also obtained. We also compare our results with those of standard approaches in which the queue selection criterion is based on the number of customers in the various queues.

A BULK SERVICE GI/M/1 QUEUE WITH SERVICE RATES DEPENDING ON SERVICE BATCH SIZE

We consider a bulk service GI/M/1 queue with service rates depending on service batch size. If there are n customers waiting at the completion of service, min(n, K) customers enter service. We show that the queue size and the service batch size at points of arrivals form an embedded Markov chain and the steady-state probabilities of this Markov chain have the matrix geometric form. We describe the rate matrix R of the matrix geometric solution procedure in a readily computable form. We obtain explicit analytic expressions for the steady-state queue length distribution at points of arrivals. Further we obtain the Laplace-Stieltjes transform and the moments of the stationary waiting time distribution of an arbitrary customer.

Analysis of a nonpreemptive priority queue with SPP arrivals of high class

This paper considers a nonpreemptive priority queueing system with two priority classes of customers, where high priority customers arrive to the system in accordance with a switched Poisson process ( SPP) and low priority customers in accordance with a Poisson process. Using the supplementary variable technique, we derive the joint probability generating function of the stationary queue length distributions and the Laplace-Stieltjes transforms of the stationary waiting time distributions of high and low priority customers. We also present some numerical results in order to show the computational feasibility of the analytical results.

Nonparametric Estimation of the Stationary Waiting Time Distribution Function for the $GI/G/1$ Queue

The $GI/G/1$ queueing model is regarded as a functional that maps the service and interarrival time distribution functions onto the stationary waiting time distribution function. By considering the output of the functional when it is applied to nonparametric estimators of the input distribution functions, we obtain a nonparametric estimator of the stationary waiting time distribution function. Using appropriate continuity and differentiability properties of the functional, we show that statistical properties of the input estimators carry over to corresponding properties for the stationary waiting time distribution function estimator.

A discrete queue with the markovian arrival process and phase type primary and secondary services

A discrete time queueing model with Markovian arrival process in which jobs require primary and possibly secondary service, is analysed. Under the assumptions of infinite waiting room for the primary queue and no waiting room for the secondary queue we show that the steady state queue length density is of the matrix-geometric type. Stationary waiting time distribution of an arriving job is derived explicitly. Efficient algorithmic procedures for the computation of the rate matrix and the waiting time distribution are given

A finite capacity queue with Markovian arrivals and two servers with group services

In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process. The system is attended by two exponential servers, who offer services in groups of varying sizes. The service rates may depend on the number of customers in service. Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady‐state queue length densities at arrivals and at arbitrary time points; (b) the Laplace‐Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. The stationary waiting time distribution is shown to be of phase type when the interarrival times are of phase type. Efficient algorithmic procedures for computing the steady‐state queue length densities and other system performance measures are discussed. A conjecture on the nature of the mean waiting time is proposed. Some illustrative numerical examples are presented.

Analysis of a finite MAP/G/1 queue with group services

The finite capacity queues, GI/PH/1/N and PH/G/1/N, in which customers are served in groups of varying sizes were recently introduced and studied in detail by the author. In this paper we consider a finite capacity queue in which arrivals are governed by a particular Markov renewal process, called a Markovian arrival process (MAP). With general service times and with the same type of service rule, we study this finite capacity queueing model in detail by obtaining explicit expressions for (a) the steady-state queue length densities at arrivals, at departures and at arbitrary time points, (b) the probability distributions of the busy period and the idle period of the server and (c) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures when services are of phase type are discussed. An illustrative numerical example is presented.

A Functional Approach to the Stationary Waiting Time and Idle Period Distributions of the GI/G/1 Queue

The GI/G/1 queueing model is regarded as a functional which maps the service and interarrival time distributions onto output quantities of interest, such as the stationary waiting time distribution. For the case where the input distributions have densities, techniques from infinite-dimensional analysis are used to obtain derivatives and Taylor series expansions for the functionals. These yield approximations to the output distributions which can be viewed as nonparametric alternatives to parametric approximations such as those provided by infinitesimal perturbation analysis or the phase method.

A finite capacity GI/PH/1 queue with group services

We consider a finite-capacity single-server queue in which arrivals occur one at a time, according to a renewal process. The successive service times are mutually independent and have a common phase-type distribution. The customers are served in groups of size at least L, a preassigned threshold value. Explicit analytic expressions for the steady-state queue-length densities at arrivals and at arbitrary time points, and the throughput of the system are obtained. The Laplace-Stieltjes transform of the stationary waiting-time distribution of an admitted customer at points of arrivals is computed. It is shown to be of phase type when the arrival process is also of phase type. Efficient algorithmic procedures for the steady-state analysis of the model are presented. These procedures are used in arriving at an optimal value for L that minimizes the mean waiting time of an admitted customer. A conjecture on the nature of the mean waiting time is proposed.

Stationary Waiting Time Distribution of a Queue in Which Customers Require a Random Number of Servers

We consider the multiserver queueing system studied originally by L. Green (1980) in which customers request service from a random number of identical servers. We provide a matrix-geometric formulation of the problem, present a simple means for computing the stationary probability vector, and propose an algorithm based on randomization for computing the waiting time distribution. We also give a numerical example and discuss issues involved in extending the formulation to include customer priority classes.

Queues with service times and interarrival times depending linearly and randomly upon waiting times

We consider a modification of the standardG/G/1 queue with unlimited waiting space and the first-in first-out discipline in which the service times and interarrival times depend linearly and randomly on the waiting times. In this model the waiting times satisfy a modified version of the classical Lindley recursion. We determine when the waiting-time distributions converge to a proper limit and we develop approximations for this steady-state limit, primarily by applying previous results of Vervaat [21] and Brandt [4] for the unrestricted recursionYn+1=CnYn+Xn. Particularly appealing for applications is a normal approximation for the stationary waiting time distribution in the case when the queue only rarely becomes empty. We also consider the problem of scheduling successive interarrival times at arrival epochs, with the objective of achieving nearly maximal throughput with nearly bounded waiting times, while making the interarrival time sequence relatively smooth. We identify policies depending linearly and deterministically upon the work in the system which meet these objectives reasonably well; with these policies the waiting times are approximately contained in a specified interval a specified fraction of time.

Exponential approximation of waiting time and queue size for queues in heavy traffic

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

Exponential approximation of waiting time and queue size for queues in heavy traffic

An exponential approximation for the stationary waiting time distribution and the stationary queue size distribution for single-server queues in heavy traffic is given for a wide class of queues. This class contains for example not only queues for which the generic sequence, i.e. the sequence of service times and interarrival times, is stationary but also such queues for which the generic sequence is asymptotically stationary in some sense. The conditions ensuring the exponential approximation of the characteristics considered in heavy traffic are expressed in terms of the invariance principle for the stationary representation of the generic sequence and its first two moments.

Moments Of the stationary waiting time in the Gi/Ph/1 queue

Recursive relations for computing the higher moments of the stationary waiting time distribution in a stable GI/PH/1 queue are derived. These provide an accurate and stable technique to compute these moments.

Moments Of the stationary waiting time in the Gi/Ph/1 queue

Recursive relations for computing the higher moments of the stationary waiting time distribution in a stable GI/PH/1 queue are derived. These provide an accurate and stable technique to compute these moments.

Transform-free equations for the stationary waiting time distributions in the queue with poisson arrivals and bulk services

We study a queue with Poisson arrivals and a bulk service rule with two thresholdsN andm on the group sizes. No group of fewer thanN or more thanm customers is served. When the number of available customers is betweenN andm, all customers are served together. The principal results deal with the joint stationary distribution of the waiting time of an arriving customer and of the size of the group in which he is eventually served. After prior computation of the stationary queue length density, the evaluation of the waiting time distribution is reduced to the solution of a system of linear differential equations and a single integral equation. The process describing the waiting time is in general non-Markovian.

On the stationary waiting-time distribution in the GI/G/1 queue, I: Transform methods and almost-phase-type distributions

This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying the GI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for the GI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.The algorithm has been coded and examples of numerical results are included.

On the stationary waiting-time distribution in the GI/G/1 queue, I: Transform methods and almost-phase-type distributions

This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying theGI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for theGI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.The algorithm has been coded and examples of numerical results are included.

Stationary waiting time distribution in queues with phase type service and in quasi-birth-and-death processes

We consider the stationary waiting time distribution under the FIFO discipline for multi-server queues with (possibly heterogeneous) phase type servers and renewal arrivals, and for queues which can be represented as a quasi-birth-and-death process. It is shown that the complementary distribution function of the stationary waiting time in these queues is of the form where θ is positive and the dj are probabilities. A complete characterization of θ and dj , which lends itself to interesting probabilistic interpretations, is given for all these cases. Except in the case of the multiserver GI/PH/c queue which is to be discussed in detail elsewhere, we provide an efficient algorithm to compute the waiting time distribution.

Algorithms for the multi-server queue with phase type service

We consider the multiserver queue with c phase type servers and a renewal input. It is well-known that the stationary joint distribution of the queue length and the number of servers in each phase of service is “modified matrix-geometric”, that is, of the form . Here, the component xv(n1,...,nm) of the vector xv is the probability that a stationary arrival finds v customers in the system and that immediately thereafter n, servers are in phase i of service, 1≤i≤m. The matrix R appears also in the formulas for all interesting queue characteristics and is therefore important. Here we develop efficient schemes to compute the matrix R and the “boundary probabilities” x0,...,xc-1. We also provide a complete proof for the easily implementable formula for the stationary waiting time distribution; this proof was sketched in an earlier paper without much detail.