Stationary Waiting Time Distribution Research Articles

A note on mixed exponential approximations for Gi/G/1 waiting times

A method is offered for the effective estimation of the stationary waiting-time distribution of the GI/G/1 queue by a (possibly nonconvex) mixed exponential CDF. The approach relies on obtaining a generalized exponential mixture as an approximation for the distribution of the service times. This is done by the adaptation of a nonlinear optimization algorithm previously developed for the maximum-likelihood estimation of parameters from mixed Weibull distributions. The approach is particularly well-suited for obtaining the delay distribution beginning from raw interarrivai and service-time data.

系GI/E_k/mにおける定常待ち分布

In this paper, the stationary waiting time distributions Fq (x) and F (x) are explicitly formulated for the GI/Ek/m queue under the first-come first-served discipline. The transition probability matrix and the imbedded probabilities play the important ro.les in this study. Some numerical results are presented for various systems as EQ/Ek/m, UQ/Ek/m, D/Ek/m, etc. Further the properties of F (x) are considered.

A Multiserver Queue with Thresholds for the Acceptance of Customers into Service

A queue with Poisson arrivals is served by N identical exponential servers. When a server becomes free, a group of customers of size at most b can be served. The server is not allowed to process a group of size less than a, 1 ≤ a ≤ b. The actual size of each group served depends on the number of waiting customers. The queue is studied as a Markov process. In the stable case, the stationary probability vector of this process has a simple, readily computable form. Using this form, the stationary waiting time distribution, which is analytically intractable, may be expressed in an algorithmically useful form. Several questions, related to the optimal design of such a service system, may be algorithmically investigated. This model serves to illustrate the advantages of the algorithmic approach. The design criteria are not analytically tractable in general.

Approximating the stationary waiting time distribution function of GI/GI/1-queues with arithmetic interarrival time and service time distribution function

This paper deals with the stationary waiting time distributionW of GI/GI/1 queues. It is restricted to the case that interarival time and service time have lattice distributions. First some estimations forW are presented. These estimations allow to construct bounds forW if an approximation ofW is given, where good approximations yield good bounds. Next we describe in detail some methods for calculation of approximations ofW. Finally we compare these methods, using a number of test problems. Diese Arbeit beschaftigt sich mit der stationaren WartezeitverteilungW von GI/GI/1 Warteschlangen. Die Darstellung beschrankt sich auf den Fall, daβ Zwischenankunftszeit und Bedienungszeit gitterformig verteilt sind. Zunachst werden Abschatzungen fur die stationare Wartezeitverteilung hergeleitet. Diese Abschatzungen gestatten es, aus einer gegebenen Naherung beidseitige Schranken furW zu berechnen. Dabei ist gesichert, daβ gute Naherungen gute Schranken liefern. Anschlieβend werden einige Moglichkeiten zur Berechnung vonW angegeben und mit Hilfe von Testbeispielen, unter Verwendung der beschriebenen Abschatzungen, verglichen.

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.

Stationary waiting-time distributions in the GI/PH/1 queue

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

Stationary waiting-time distributions in the GI/PH/1 queue

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

The effect of variability in the GI/G/s queue

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

The effect of variability in the GI/G/s queue

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

Approximations of the Mean Waiting Time in an M/G/s Queueing System

This paper considers the problem of obtaining approximate expressions for the first moment WGs of the stationary waiting time distribution in an M/G/s queueing system. Special attention is paid to the case G ≡ D, i.e., constant service times. Most known approximations are in fact heavy traffic approximations which have rather large relative errors in the light traffic case. In the present study both the light traffic and heavy traffic behavior of WGs (WDs) are taken into account. In order to obtain mean waiting time approximations it appears to be useful to introduce a quantity (the “normed cooperation coefficient”) which is inversely proportional to WGs and which is in some sense a measure for the “cooperation” between the servers of the service facility. A part of the paper is devoted to the analysis of this normed cooperation coefficient.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

Note—A Single Server Queue with Constant Service Time and Restricted Accessibility

A queuing system has restricted accessibility if not every customer is admitted to the system. In this paper a single server queuing system is studied, where the arriving customers are rejected if their waiting plus service times would exceed a fixed amount K. The arrivals follow a Poisson process. In this case an equation for the stationary waiting time distribution is given. The explicit solution of this equation is obtained, when the service time is assumed to be constant. For this model the queue length distribution is also easily derived. It is observed that the queue with finite waiting room is a special case of this model.

Level Crossings in Point Processes Applied to Queues: Single-Server Case

This paper introduces a new methodology for obtaining the stationary waiting time distribution in single-server queues with Poisson arrivals. The basis of the method is the observation that the stationary density of the virtual waiting time can be interpreted as the long-run average rate of downcrossings of a level in a stochastic point process. Equating the total long-run average rates of downcrossings and upcrossings of a level then yields an integral equation for the waiting time density function, which is usually both a linear Volterra and a renewal-type integral equation. A technique for deriving and solving such equations is illustrated by means of detailed examples.

Further stochastic order relations among GI/GI/1 queues with a common traffic intensity

Using two relations between virtual and actual waiting time distribution, it is shown that the stationary waiting times in GI/GI/1 queues, where both service and inter-arrival time distributions are of type 1SFBUE, are stochastically smaller than in M/M/1 with the same traffic intensity. Furthermore, a relation between virtual waiting time and distribution of number of customers m system is discussed.Finally, using a result of Kollerstrom a bound for the stationary waiting time distribution being a rational function is given, and it is shown that the coefficient of variation of stationary waiting times is greater than one.

On exponential bounds for the waiting-time distribution function in GI/G/1

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

On exponential bounds for the waiting-time distribution function in GI/G/1

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

Erhaltungssätze im wartemodell g/g/l mit "erwärmung"

A generalisation of the waiting model GiG/l is considered: The first order in every busy period suffers a random delay with distribution function Cbefore its service commences. As according to [10] the stationary waiting time distribution W′ results from the stationary waiting time W of the fundamental model G/G/1 by W′ = W * G, connections between C and G are studied. By making a special assumption the characteristic function corresponding to G is analytical or rrtioncl if and only if the characteristic function corresponaing to C is analytical or rational. I n addition to that G is a mixture of negtive-exponential random variables if and only if C is a mixture of this manner.

Die bedeutung von KINGMANS integralungleichnngen bei der approximation der statlonären wartezeitverteilung im modell gi/g/1 mit und ohne verzögerung beim beginn einer beschäftigungsperiode

We consider the stationary waiting-time distribution W in GI/G/1. KINGMAN [4] used two integral inequalities to obtain estimations for W above and below. But he gave only one solution explicitely for each of them. We show that the inequalities possess a great methodological importance. By means of them we first get a result of DALEY-MORAN [1] which permits to find further solutions. But the very point is a methed resting on the inequalities to improve every solutions successively. Examples for the refinement of KINGMANS estimations are given in (1.11)-(1.13). - The method can be carried over to a generalization of GI/G/1 in which a custumer who finds the server idle is not served immediately but only after a randam "warming- up time".

Simulating Stable Stochastic Systems, I: General Multiserver Queues

A technique is introduced for analyzing simulations of stochastic systems in the steady state. From the viewpoint of classical statistics, questions of simulation run duration and of starting and stopping simulations are addressed. This is possible because of the existence of a random grouping of observations which produces independent identically distributed blocks from the start of the simulation. The analysis is presented in the context of the general multiserver queue, with arbitrarily distributed interarrival and service times. In this case, it is the busy period structure of the system which produces the grouping mentioned above. Numerical illustrations are given for the M/M/1 queue. Statistical methods are employed so as to obtain confidence intervals for a variety of parameters of interest, such as the expected value of the stationary customer waiting time, the expected value of a function of the stationary waiting time, the expected number of customers served and length of a busy cycle, the tail of the stationary waiting time distribution, and the standard deviation of the stationary waiting time. Consideration is also given to determining system sensitivity to errors and uncertainty in the input parameters.