Perturbation analysis of waiting times in the G/G/1 queue

This paper is devoted to perturbation analysis of the stationary distribution of waiting times in the G/G/1 queue with a parameter-dependent service time distribution. We provide sufficient conditions under which the stationary distribution is Lipschitz continuous and we explicitly compute the Lipschitz constant. Thereby, we provide bounds on the effect of a (finite) perturbation of the service time distribution on the stationary waiting time. The case of infinitesimal perturbations (read, derivatives) is treated as well.     

A finite capacity BMAP K /G K /1 queue with the generalized foreground-background processor-sharing discipline

A queueing system with a batch Markov arrival process, several types of customers, generalized foreground-background processor-sharing discipline with minimal served length, and separate finite buffers for customers of different types or a common finite buffer for customers of all types is studied. Mathematical relations for computing the stationary joint distributions of the number of customers of all types in the system are derived.     

Approximations for the waiting time distribution of the M/G/c queue

For the M/G/c queue we present an approximate analysis of the waiting time distribution. The result is given in the form of the defective renewal equation. This integral equation can be numerically solved by a simple recursion procedure. Also, asymptotic results for the waiting times are presented. Numerical results indicate that the approximations are sufficiently accurate for practical purposes.     

A polynomial factorization approach to the discrete time GI/G/1/(N) queue size distribution

The queue of a single server is considered with independent and identically distributed interarrivai and service times and an infinite (GI/G/1) or finite (GI/G/1/N) waiting room. The queue discipline is non-preemptive and independent of the service times.

A discrete time version of the system is analyzed, using a two-component state model at the arrival and departure instants of customers. The equilibrium equations are solved by a polynomial factorization method. The steady state distribution of the queue size is then represented as a linear combination of geometrical series, whose parameters are evaluated by closed formulae depending on the roots of a characteristic polynomial.

Considering modified boundary constraints, systems with finite waiting room or with an exceptional first service in each busy period are included.     

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size $m$ and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order $m$. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order $m$, but no derivation for the queue length was provided in the general case.This paper introduces an order $m^2$ phase-type representation $(\kappa ,K)$ for the queue length distribution in the general case and proves that the order $m^2$ of the distribution cannot be further reduced in general. A matrix geometric representation $(\kappa ,K,\nu )$ is also established for the number of type $\tau ?\{1,\dots ,m\}$ customers in the system, where a customer is of type $\tau$ if the phase in which it completes service belongs to $\tau$. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute $(\kappa ,K,\nu )$. When the arrivals have a Markovian structure, the numerical procedure is reduced to solving a Quasi–Birth–Death (for the discrete time case) or fluid queue (for the continuous time case).Finally, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order $m$ phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the $R$ matrix of a Quasi–Birth–Death Markov chain.     

A new procedure to estimate waiting time in GI/G/2 system by server observation

Most GI/G/2 queueing formulae need the variance of inter-arrival time, which is in many cases more difficult to estimate than the other values used in the formulae such as the mean of inter-arrival time, mean of service time and variance of service time. This paper presents a new GI/G/2 queueing formula which uses a slightly different set of data easier to obtain than the variance of inter-arrival time. The key variables are the numbers of system busy periods and system idle periods. Also, it is shown, by simulation, that the waiting time estimation error from the new formula is far less than other popular queueing formulae which use the first two moments of service time and inter-arrival time over a wide range of coefficient of variation.Scope and purposeWaiting is very common in our daily life, and the estimation is sometimes very important for the design of service and manufacturing systems. If the number of barbers at a barber's shop is too small, customers frequently wait for the service too long. If the number of machines in a manufacturing shop is too small, the production lead time from order entry to product delivery can be very long.The waiting time is closely related not only to the average service requirement but also to the variability of it. If customers require service at the same time, the average waiting time of the customers will be longer than the average waiting time with even requests. Traditionally, the variance of inter-arrival time has been used to represent the variability; however, estimation of the variance needs observation of customer arrivals, which often needs much effort. This paper presents another procedure to estimate the waiting time. This procedure does not need the observation of customers. The estimation of waiting time for bank teller machines can be a good application example of this new procedure because the machines do not have the arrival data of the customers. The procedure presented here is for a two parallel server case.     

Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for multi-class egalitarian PS queues, we show that the marginal queue length distribution for each class equals the queue length distribution of an equivalent single class PS model with a random number of permanent customers. Similarly, the mean sojourn time (conditioned on the initial service requirement) for each class can be obtained by conditioning on the number of permanent customers. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian PS models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights.     

Maximum entropy and the G/G/1/N queue

Summary A new hybrid analytic framework, based on the principle of maximum entropy, is used to derive a closed form expression for the queue length distribution of a G/G/1 finite capacity queue. It is shown that Birth-Death homogeneous recursions for a single resource queue are special cases of maximum entropy one-step transitions which can be applied either in an operational or stochastic context. Furthermore, an equivalence relationship is used to analyse two-stage cyclic queueing networks with general service times, and favourable comparisons are made with global balance and approximate results. Numerical examples provide useful information on how critically system behaviour is affected by the distributional form of interarrival and service patterns. Comments on the implication of the work to the performance analysis and aggregation of computer systems are included.Some of the material included in this paper has been presented to the Performance '86 and ACM Sigmetrics 1986 Joint Conference on Computer Modelling, Measurement and Evaluation, May 28–30, 1986, University of North Carolina, USA     

Distribution of virtual and actual waiting time in Hr/G/1 and Er/G/1 queueing systems

A relationship between the stationary distribution of customer waiting time and virtual waiting time in the k-th arrival stage is derived for H_r/G/1 and E_r/G/1 queueing systems. The Laplace transforms of these distributions are obtained for the H_r/G/1 system.Translated from Kibernetika, No. 5, pp. 94–97, September–October, 1989.     

The complete analysis of the discrete time finite DBMAP/G/1/N queue

A comprehensive analysis of the DBMAP/G/1/N queueing system for both the “arrival first” and “departure first” policies is given including detailed proofs and latest results on an algorithmic recipe to compute the z-transform of the waiting time probability function as well as the probability function itself. Additionally, expressions for the probability of simultaneous arrivals and departures and an algorithm to compute them are presented for the first time. The framework presented allows dealing with a superposition DMAP+DBMAP as an input process. For both streams in the superposition, per stream waiting time probability functions are given for the case of a deterministic service time, which is relevant for the modeling of ATM networking (both wired and wireless). Together with findings for the continuous time case, this paper completes the insights into finite queueing systems of the M/G/1-type. In comparison with the method of the unfinished work, which can be used to analyze finite queueing systems with deterministic service-time only and, besides other semi-Markovian input processes, with the DBMAP as an input, the M/G/1-paradigm provides much faster algorithms to compute loss probabilities and waiting time moments, due to the smaller system matrix. Note that the D[B]MAP has proved a versatile stochastic process, which can also be tuned to represent periodic correlation functions and not only geometrically decaying ones.     

Moments of the queue size distribution in the MAP/G/1 retrial queue

We consider an MAP/G/1 retrial queue. A necessary and sufficient condition is obtained for the existence of the moments of the queue size distribution. The condition is expressed in terms of the moment condition for a service time distribution. In addition, we provide recursive formulas for the moments of the queue size distribution. Numerical examples are given to illustrate our results.     

A functional approximation for the M/G/1/N queue

This paper presents a new approach to the functional approximation of the M/G/1/N built on a Taylor series approach. Specifically, we establish an approximative expression for the remainder term of the Taylor series that can be computed in an efficient manner. As we will illustrate with numerical examples, the resulting Taylor series approximation turns out to be of practical value.