Analysis of a MAP/M/1/N Queue with Periodic and Non-Periodic Piecewise Constant Input Rate (original) (raw)

Transient Behavior of the MAP/M/1/N Queuing System

Mathematics, 2021

This paper investigates the characteristics of the MAP/M/1/N queuing system in the transient mode. The matrix method for solving the Kolmogorov equations is proposed. This method makes it possible, in general, to obtain the main characteristics of the considered queuing system in a non-stationary mode: the probability of losses, the time of the transient mode, the throughput, and the number of customers in the system at time t. The developed method is illustrated by numerical calculations of the characteristics of the MAP/M/1/3 system in the transient mode.

Analysis of MAP/M/2/K queueing model with infinite resequencing buffer

Performance Evaluation, 1998

We consider a MAP/Mj2/K queueing model in which messages should leave the system in the order in which they entered into the system. In the case of infinite resequencing buffer, the steady-state probability vector is shown to be of matrix-geometric type. The total sojourn time of an admitted message into the system is shown to be of phase type. Efficient algorithmic procedure:; for computing various performance measures are given, and some interesting numerical examples are discussed. 0 1998 Elsl:vier Science B.V.

The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates

2021

We study a queueing system with Erlang arrivals with k phases and Erlang service with m phases. Transition rates among phases vary periodically with time. For these systems, we derive the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series.

Numerical Solution of Piecewise-Stationary M t / G t /1 Queues

Operations Research, 1997

We develop an algorithm for computing the (exact) cumulative distribution function of the timedependent workload in a piecewise-stationary M t / G t / 1 queue with a work-conserving service discipline and general service-time distributions, where service times are determined at arrival instants. The t subscripts indicate that the arrival rate and the general service-time distribution may change with time, but we allow changes only at finitely many time points. The algorithm is based on numerical transform inversion, using the classical Taka ´ cs double-transform of the transient workload in an M/G/1 queue recursively over the successive stationary intervals. In particular, we apply our recently developed Fourier-series-based inversion algorithms for two-dimensional transforms and nested one-dimensional transforms. We also do additional work to greatly speed up the computation while tightly controlling the error. As a consequence, the computation time grows only quadratically with the number of intervals.

A Note on the Waiting-Time Distribution in an Infinite-Buffer GI[X]/C-MSP/1 Queueing System

Journal of Probability and Statistics

This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (C-MSP). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution....

Transient Analysis of Queueing Model

Journal of the Institute of Engineering, 2016

This paper deals with the study of Erlangian queueing system with time dependent framework. Under our study we find (i) the expected number of customers in the queue (ii) the expected waiting time before being served (iii) the expected time spent in the system (iv) the expected number of customers in the system. Customers arrive in the system in Poisson fashion with rate and served in arbitrary service time distribution with rate µ.The probability generating function technique and Laplace transform method have been used. The numerical computation has also been obtained for applicability of the model.

The busy period for the M0/G/1/m system with service time dependent of the queue length

Journal of Applied Mathematics and Computational Mechanics, 2013

We consider the M θ /G/1/m system wherein the service time depends on the queue length and it is determined at the beginning of customer service. Using an approach based on the potential method proposed by V. Korolyuk, the Laplace transforms for the distribution of the number of customers in the system on the busy period and for the distribution function of the busy period are found.

Steady-state analysis of single exponential vacation in a PH/MSP/1/∞ queue using roots

2017

We consider an infinite-buffer single-server queue where inter-arrival times are phase-type (PH), the service is provided according to Markovian service process (MSP ), and the server may take single, exponentially distributed vacations when the queue is empty. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function (VGF) of system-length distribution at a pre-arrival epoch. Also, we obtain the steady-state system-length distribution at an arbitrary epoch along with some important performance measures such as the mean number of customers in the system and the mean system sojourn time of a customer. Later, we have established heavyand light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. At the end, we present numerical results in the form of tables to show the effect of model parameters on the performance measures.

Waiting time distribution in a class of discrete-time cyclic service multi-queue systems

Performance Evaluation, 1997

A new approximate method is developed for finding the waiting and sojourn time distributions in a class of multi-queue systems served in cyclic order at discrete intervals. An immediate application for such a model is in communication networks where a number of different traffic sources compete to access a group of transmission channels operating under a time-slotted sharing policy. This system maps naturally onto a model in which the inter-visit time has a probability mass function of phase-type. We derive a set of matrix equations with easily tractable iterative procedures for their solution and controllable accuracy in their numerical evaluation. We then validate the analytical model against simulation and discuss the validity of the assumptions. This methodology can be extended to several other polling strategies. © 1997 Elsevier Science B.V. a stochastic decomposition of the system's unfinished work. E:cact expressions for weighted sums of mean waiting times were derived using this method (see [i]). Several other methods have been developed for computing the mean delay, the mean queue length for each queue, the amount of work of the server and the cycle time. A survey of these methods is given by Levy and Sidi (see [ 10]). This paper tackles the problem of finding an approximate method for evaluating the waiting time pi ,'Jability density function in multi-queue systems with discrete service time. More abstractly, it also considers the sojourn ~.ime distribution of systems with exceptional first service times. A direct application for such systems can be found in time-slotted medium access protocols. In these systems several traffic sources compete to use a group of transmission channels. These transmitters are available only at the beginning of fixed length time-slots. At each discrete interval a single server starts visiting queues in cyclic order and assigning packets to the transmitters.