Javad Tavakoli

A Tandem Queue with a Movable Server: An Eigenvalue Approach

In this paper, we analyze a two station tandem queue with Poisson arrivals and exponential service times. All arrivals occur at the first line, and, after receiving service at the first station, they proceed to the second line. There is only a finite buffer between the stations, and, as soon as the buffer is full, any job completed by the first server is lost. To reduce customer loss, the first server can move to the second station and help the second server, thereby increasing its service rate. Once the work at station two is complete, the job leaves the system. The problem will be solved by using eigenvalues which can be obtained in explicit form. It is shown that this method is substantially faster than matrix analytic methods.

Analysis of exact tail asymptotics for singular random walks in the quarter plane

In this paper, we consider all singular cases of random walks in the quarter plane. Specifically, exact light tail asymptotics for stationary probabilities are obtained for all singular random walks.

Optimal policies of M(t)/M/c/c queues with two different levels of servers

This paper deals with optimal control points of M(t)/M/c/c queues with periodic arrival rates and two levels of the number of servers. We use the results of this model to build a Markov decision process (MDP). The problem arose from a case study in the Kelowna General Hospital (KGH). The KGH uses surge beds when the emergency room is overcrowded which results in having two levels for the number of the beds. The objective is to minimize a cost function. The findings of this work are not limited to the healthcare; They may be used in any stochastic system with fluctuation in arrival rates and/or two levels of the number of servers, i.e., call centers, transportation, and internet services. We model the situation and define a cost function which needs to be minimized. In order to find the cost function we need transient solutions of the M(t)/M/c/c queue. We modify the fourth-order Runge–Kutta to calculate the transient solutions and we obtain better solutions than the existing Runge–Kutta method. We show that the periodic variation of arrival rates makes the control policies time-dependent and periodic. We also study how fast the policies converge to a periodic pattern and obtain a criterion for independence of policies in two sequential cycles.

Transient solutions for multi-server queues with finite buffers

Transient solutions for M/M/c queues are important for staffing call centers, police stations, hospitals and similar institutions. In this paper we show how to find transient solutions for M/M/c queues with finite buffers by using eigenvalues and eigenvectors. To find the eigenvalues, we create a system of difference equations where the coefficients depend on a parameter x. These difference equations allow us to search for all eigenvalues by changing x. To facilitate the search, we use Sturm sequences for locating the eigenvalues. We also show that the resulting method is numerically stable.

Comparing Some Algorithms for Solving QBD Processes Exhibiting Special Structures

Abstract The objective of this paper is to find equilibrium probabilities of Quasi-Birth-Death (QBD) processes with a special structure. A QBD process is a Markov chain in which the states are divided into levels, with each level containing several phases. Moreover, only transitions between neighboring levels are allowed, that is, no event can increase or decrease the level by more than 1. Transitions between levels are governed by three matrices: the rates of going up one level are given by the matrix A0, the rates of staying at the same level by the matrix A1, and the rates of going down one level by A2. QBD processes are frequently used to model queueing systems, with the level often being the number in some queue, and in many queueing situations, one can go up one level only when in specific phases, and it is easy to see that the only rows of A0 containing non-zero elements are the rows corresponding to these phases. Hence, we investigate the case where A0 has only a small number of rows which are not zero, and we show how this can be exploited in algorithm T, an algorithm due to Neuts, and in algorithm U suggested by Latouche. In many cases, A0 is lower triangular, and this can be exploited by algorithm U, but not by algorithm T. There is also a logarithmic reduction method, but unfortunately, we have not found any way to significantly reduce the computational complexity of this method when A0 has vanishing rows, or A0 is lower triangular. We also show how to deal with the case where A0 does not have the full rank, but few or no rows of zero.

TWO-STATION QUEUEING NETWORKS WITH MOVING SERVERS, BLOCKING, AND CUSTOMER LOSS ∗

This paper considers a rather generalmodelinvol ving two exponentialservers, each having its own line. The first line is unlimited, whereas the second line can only accommodate a finite number of customers. Arrivals are Poisson, and they can join either line, and once finished, they can either leave the system, or they can join the other line. Since the space for the second line is limited, some rules are needed to decide what happens if line 2 is full. Two possibilities are considered here: either the customer leaves prematurely, or he blocks the first server. The model also has moving servers, that is, the server at either station, while idle, can move to help the server of the other station. This model will be solved by an eigenvalue method. These eigenvalue methods may also prove valuable in other contexts.

The continuous spectrum for the M / M / c queue

Transient solutions for M/M/c queues are important for many purposes, in particular for staffing facilities such as call centers. In this article, we show how to use spectral analysis to find such solutions. The difficulty is that unless the number in line is bounded, one has to deal with matrices of infinite size, and hence with a countable infinite number of eigenvalues. This problem can be overcome by noting that the spectrum is dense with few exceptions. We also show how many discrete eigenvalues remain. Our theory may also work to obtain spectra for other infinite-dimensional matrices. Numerical properties of our approach are explored.

A bayesian approach to find random-time probabilities from embedded markov chain probabilities

The embedded Markov chain approach is widely used in queuing theory, in particular in M/G/1 and GI/M/c queues. In these cases, one has to relate the embedded equilibrium probablities to the corresponding random-time probabilities. The classical method to do this is based on Markov renewal theory, a rather complex approach, especially if the population is finite or if there is balking. In this article we present a much simpler method to derive the random-time probabilities from the embedded Markov chain probabilities. The method is based on conditional probability. Our approach might also be applicable in such situations.

Stochastic and substochastic solutions for infinite-state Markov chains with applications to matrix-analytic methods

This paper deals with censoring of infinite-state banded Markov chains. Censoring involves reducing the time spent in states outside a certain set of states to 0 without affecting the number of visits within this set. We show that, if all states are transient, there is, besides the standard censored Markov chain, a nonstandard censored Markov chain which is stochastic. Both the stochastic and the substochastic solutions are found by censoring a sequence of finite transition matrices. If all matrices in the sequence are stochastic, the stochastic solution arises in the limit, whereas the substochastic solution arises if the matrices in the sequence are substochastic. We also show that, if the Markov chain is recurrent, the only solution is the stochastic solution. Censoring is particularly fruitful when applied to quasi-birth-and-death (QBD) processes. It turns out that key matrices in such processes are not unique, a fact that has been observed by several authors. We note that the stochastic solution is important for the analysis of finite queues.

Locally free vector spaces in a topos

In this work it is shown that in an elementary topos with natural number object and with internal choice, every vector space over a field is locally free.