Steady-state Probability Vector Research Articles

Analysis of a retrial queuing model with MAP arrivals and two types of customers

In this paper, we consider a queuing model in which two types of customers, say, Types 1 and 2, arrive according to a Markovian arrival process (MAP). Type 1 customers have a buffer of capacity K and are served in groups of varying sizes ranging from a predetermined value L to a maximum size, K. The service times are exponentially distributed. Type 2 customers are served one at a time and the service times are assumed to be exponential. The system has two servers, of which one is totally dedicated to serving Type 2 customers. The other server can serve both Types 1 and 2 customers. Any arriving Type 1 customer finding the buffer full is considered lost. Any Type 2 customer not entering into service immediately orbit in an infinite space. These orbiting customers compete for service by sending out signals that are exponentially distributed. The steady state probability vector of this queuing model is of matrix-geometric type with a highly sparse rate matrix. This sparsity is exploited in the analysis and several interesting numerical examples are discussed. The Laplace-Stieltjes transform LST of the waiting time distribution of a Type 1 customer at an arrival epoch is derived.

Maximum entropy solution to a quorum queueing system

For a quorum queueing system, the system size steady-state initial probability vector is usually found by solving a simple system of linear equations, but computation of the system size steady-state tail probability vector requires the inversion of the probability generating function. We use in this paper the initial probability vector to compute the mean system size and then, using the mean system size, we compute the maximum entropy approximation of the tail probability vector, thus avoiding inversion of the probability generating function. In a numerical example, comparison of the initial probability vectors obtained by the exact and approximate methods proved to be excellent. We also show how to compute the tail probability vector.

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

In this paper we deal with iterative numerical methods to solve linear systems arising in continuous-time Markov chain (CTMC) models. We develop an algorithm to dynamically tune the relaxation parameter of the successive over-relaxation method. We give a sufficient condition for the Gauss–Seidel method to converge when computing the steady-state probability vector of a finite irreducible CTMC, and a sufficient condition for the generalized minimal residual projection method not to converge to the trivial solution 0 when computing that vector. Finally, we compare several splitting-based iterative methods and a variant of the generalized minimal residual projection method.

Stochastic complement analysis of multi-server threshold queues with hysteresis

We consider a K-server threshold-based queueing system with hysteresis, for which a set of forward thresholds ( F 1, F 2,…, F K−1 ) and a set of reverse thresholds ( R 1, R 2,…, R K−1 ) are defined. A simple version of this multi-server queueing system behaves as follows. When a customer arrives to an empty system, it is serviced by a single server. Whenever the number of customers exceeds a forward threshold F i , a server is added to the system and server activation is instantaneous. Whenever the number of customer falls below a reverse threshold R i , a server is removed from the system. We consider and solve several variations of this problem, namely: (1) homogeneous servers with Poisson arrivals, (2) homogeneous servers with bulk (Poisson) arrivals, and (3) heterogeneous servers with Poisson arrivals. We place no restrictions on the number of servers or the bulk sizes or the size of the waiting room. In [O.C. Ibe, J. Keilson, Multi-server threshold queues with hysteresis, Performance Evaluation 21 (1995) 185–212], the authors solve a limited form of this problem using Green’s function method. More specifically, they give a closed-form solution for a K-server system, when the servers are homogeneous, and for a two-server system, when the servers are heterogeneous; the authors experienced difficulties in extending Green’s function method beyond the case of two heterogeneous servers. Rather than using Green’s function, we solve this problem using the concept of stochastic complementation, which is a more intuitive and more easily extensible method. For the case of a homogeneous multi-server system we are able to derive a closed-form solution for the steady state probability vector; for the remaining cases we give an algorithmic solution. Note, however, that we can use stochastic complementation to derive closed-form solutions for some limited forms of cases (2) and (3), such as heterogeneous servers with K=2 and bulk arrivals with a limited bulk size. Finally, our technique works both for systems with finite and infinite waiting rooms.

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

We consider a MAP/M/2/ 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 procedures for computing various performance measures are given, and some interesting numerical examples are discussed.

Near Complete Decomposability: Bounding the Error by a Stochastic Comparison Method

An aggregation technique of ‘near complete decomposable' Markovian systems has been proposed by Courtois [3]. It is an approximate method in many cases, except for some queuing networks, so the error between the exact and the approximate solution is an important problem. We know that the error is O(ε), where ε is defined as the maximum coupling between aggregates. Some authors developed techniques to obtain a O(ε k) error with k > 1 error with k > 1, while others developed a technique called ‘bounded aggregation’. All these techniques use linear algebra tools and do not utilize the fact that the steady-state probability vector represents the distribution of a random variable. In this work we propose a stochastic approach and we give a method to obtain stochastic bounds on all possible Markovian approximations of the two main dynamics: short-term and long-term dynamics.

Near Complete Decomposability: Bounding the Error by a Stochastic Comparison Method

An aggregation technique of ‘near complete decomposable' Markovian systems has been proposed by Courtois [3]. It is an approximate method in many cases, except for some queuing networks, so the error between the exact and the approximate solution is an important problem. We know that the error is O(ε), where ε is defined as the maximum coupling between aggregates. Some authors developed techniques to obtain a O(ε k ) error with k > 1 error with k > 1, while others developed a technique called ‘bounded aggregation’. All these techniques use linear algebra tools and do not utilize the fact that the steady-state probability vector represents the distribution of a random variable. In this work we propose a stochastic approach and we give a method to obtain stochastic bounds on all possible Markovian approximations of the two main dynamics: short-term and long-term dynamics.

Discrete NT-policy single server queue with Markovian arrival process and phase type service

We consider a discrete time single server queueing system in which arrivals are governed by the Markovian arrival process. During a service period, all customers are served exhaustively. The server goes on vacation as soon as he/she completes service and the system is empty. Termination of the vacation period is controlled by two threshold parameters N and T, i.e. the server terminates his/her vacation as soon as the number waiting reaches N or the waiting time of the leading customer reaches T units. The steady state probability vector is shown to be of matrix-geometric type. The average queue length and the probability that the server is on vacation (or idle) are obtained. We also derive the steady state distribution of the waiting time at arrivals and show that the vacation period distribution is of phase type.

Monotone iterative methods for Markov chains

This paper deals with monotone iterative methods for the computation of the steady-state probability vector of irreducible Markov chains. The emphasis is laid on verification techniques aiming at deriving, from an approximation, true bounds on the solution.

A discrete single server queue with Markovian arrivals and phase type group services

We consider a single‐server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady‐state probability vector is shown to be of (modified) matrix‐geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady‐state probability vector, and some important system performance measures are developed. The steady‐state waiting time distribution is derived explicitly. Some numerical examples are presented.

A general bulk service queue with arrival rate dependent on server breakdowns

A single-server queueing system with general bulk service that alternates stochastically between operational state and failed (or repair) state is considered. The system, when operational, functions as a single-server Poisson queue with general bulk service. When it fails, no service occurs but customers continue to arrive according to a Poisson process; however, the arrival rate is different from that when the system is operational. The durations of the operating periods and repair periods follow exponential and phase-type distributions, respectively. The steady-state probability vector of the number of customers in the queue and the stability condition are obtained using a matrix-geometric algorithmic approach.

Two units connected in series with general bulk service of accessible batches in unit 2

We study a queueing model consisting of two units connected in series with a finite intermediate waiting room. The customers in the buffer are served according to a new general bulk service rule with exponential times which admits each batch served to have not less than a and not more than b customers such that the arriving customers can enter service, without affecting the service time, if the size of the batch under service is less than b. For this model we obtain the steady-state probability vector for the number of customers in the queue using a matrix-geometric algorithmic approach.

A multiserver queue with markovian arrivals and group services with thresholds

We consider a multiserver queueing system in which arrivals are governed by a Markovian arrival process. The system is attended by K identical exponential servers. Under a dynamic probabilistic service rule which depends on two threshold parameters, this model is studied as a Markov process. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the steady-state probability vector and some key performance measures of the system are developed. Some numerical examples are discussed. © 1993 John Wiley & Sons, Inc.

A nonpreemptive priority multiserver queueing system with general bulk service and heterogeneous arrivals

In this paper a multiserver nonpreemptive priority queueing system consisting of two types of customer units with Poisson arrival and exponential service time distribution is analysed. The lower priority units are served in batches according to a general bulk service rule. The higher priority units have nonpreemptive priority over lower priority units. The higher priority units are served singly. The steady state probability vectors of the number of customers in the queue are obtained using the modified matrix-geometric method. The stability condition and mean queue length of customers are obtained. Numerical results are also presented.

Tandem queue with three multiserver units and bulk service in unit III

In this paper a queueing model consisting of three units in series, with a finite number of parallel servers at each unit, is considered. Buffers of nonidentical finite capacity are available between the units. A buffer of infinite capacity is assumed to exist in front of the first unit. The occurrence times for all events have negative exponential distributions. Customers in the units I and II are served singly. Customers in the buffer zone between units II and III are served according to the general bulk service rule. The service times in the ith unit ( i = 1, 2, 3) are assumed to be independently identically distributed random variables with negative exponential distributions with parameters μ i ( i = 1, 2, 3). The steady state probability vector of the number of customers in the queue is obtained using a modified matrix-geometric method. The stability condition is also obtained.

Queues with two units connected in series with a multiserver general bulk service in unit II

This paper analyses a queueing model consisting of two units I and II connected in series, separated by a finite buffer of size N. Unit I has only one exponential server capable of serving customers one at a time. Unit II consists of c parallel exponential servers and they serve customers in groups according to the general bulk service rule. The steady state probability vector of the number of customers waiting and receiving service in unit I and waiting in the buffer is obtained using the modified matrix-geometric method. Numerical results are also presented.

A tandem queue with general bulk service and servers' “vacation”

This paper analyses two queueing models consisting of two units I and II connected in series, separated by a finite buffer of size N. In both models, unit I has only one exponential server capable of serving customers one at a time and unit II consist of c parallel exponential servers, each of them serving customers in groups according to general bulk service rules. When the queue length in front of unit II is less than the minimum of batch size, the free servers take a vacation. On return from vacation, if the queue length is less than the minimum, they leave for another vacation in the first model, whereas in the second model they wait in the system until they get the minimum number of customers and then start servicing. The steady-state probability vector of the number of customers waiting and receiving service in unit I and waiting in the buffer is obtained for both the models, using the modified matrix geometric method. Numerical results are also presented.

Using state reduction for computing steady state vectors in Markov chains ofM/G/1 type

This paper presents a state reduction based algorithm for computing the steady state probability vectors of the embedded Markov chains ofM/G/1 type. The algorithm is based on the use of the notion of a state reduction box for streamlining compution. The computational details are linked directly to the theoretical results developed recently by Grassmann and Heyman [6]. Exploiting these connections and a method given in Neuts [18] for finding theG matrix forPH/PH/1 queues, we also propose an hybrid approach for solvingPH/PH/1 queues. Using several numerical examples, we report our computational experiences and present some observations about the relative merits of these approaches.

Iterative algorithms for large stochastic matrices

Markov chains have always constituted an efficient tool to model discrete systems. Many performance criteria for discrete systems can be derived from the steady-state probability vector of the associated Markov chain. However, the large size of the state space of the Markov chain often allows this vector to be determined by iterative methods only. Various iterative methods exist, but none can be proved a priori to be the best. In this paper, we propose a practical measure which allows the convergence rate of the various existing methods to be compared. This measure is an approximation of the modulus of the second largest eigenvalue of the iteration matrix and can be determined a priori. The model of a queueing network is used as an example to compare the convergence of several iterative methods and to show the accuracy of the measure.

Moment analysis for traffic associated with Markovian queueing systems

A study is made of various traffic processes associated with Markovian queuing systems, and the moments of the traffic processes are analyzed. The purpose of the study is to develop a means of computing the moments efficiently so as to extend the conventional method of representing the traffic processes to a larger class of queuing systems. As an outcome of the analysis, the recurrence formulae for computing the moments including cross-covariances between multiple traffic processes are derived and expressed in terms of the infinitesimal generator, steady-state probability vector of the Markovian queuing system and a matrix describing traffic rate. The main feature of the analysis is in the utilization of the special block-partitioned structure of the matrices involved. The results are of wide application for the study of various telecommunication networks. Two examples of applications for the Markovian queuing model with batch arrivals and a multislot integrated services system are presented. >