The N threshold policy for the GI/M/1 queue

Queueing Systems - Theory and Applications, 2010

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ C

Performance Evaluation, 2007

We consider a discrete time single server queue with discrete autoregressive process of order 1 (DAR(1)) input. By extracting a Markov process from the queue size process and applying the BASTA property, we derive simple recursive formulae for the stationary distributions of the queue size and the waiting time. These formulae are simple, numerically stable and transformfree. A stochastic decomposition property is given for the stationary waiting time, and relations between the distributions of the stationary queue size and the stationary waiting time are discussed. Numerical examples are given for stationary distributions of the queue size and the waiting time for various DAR(1) inputs.

The paper studies the queuing model with three non-identical exponential servers S 1 , S 2 and S 3 and provides a matrix-geometric solution for an underlying quasi birth-and-death (QBD) queue of an M/M(S 1),M(S 2 ,S 3)/3/(m,∞) system. Customers arrive individually according to a Poisson process and form two parallel queues, say q 1 and q 2. The size of q 1 represents the system length (queue+server) of a finite queueing facility M/M(S 1)/1/(m+1) and the size of the q 2 accounts the system length (queue+server) of a two-server queue M/M(S 2 ,S 3)/2 facility. Queue management for each of q 1 and q 2 is through a 'First Come First Served (FCFS)' basis but according to the norms of an m-policy. At an arrival instant, if the size of q 1 is strictly less than 'm', the new arrival is assigned to q 1 with an unknown probability P 1 (=1-Pr(q 1 =m)); otherwise it is assigned to q 2 with probability (1-P 1) subject to a condition that switching from q 1 to q 2 and vice versa is to be avoided. At every service completion epoch, the dispatching mechanism of the m-policy either assigns a customer of q 1 > 0 to server S 1 or a customer of q 2 >0 to server S 2 , if available, or otherwise to S 3. The underlying QBD process representing the number of customers in the system under study is formulated as a bi-variate queue length sequence X=(q 1 = i, q 2 = j) defined on the two-dimensional state space Ω ={(i, j): 0≤ i ≤ m, j ≥ 0}. Explicit expressions for the stationary condition, stationary distribution of X, marginal expected values of q 1 and q 2 , and the probability P 1 are obtained. The paper also constructs a formal linear programming to find an optimal value of m, corresponding to the minimum cost.

OPSEARCH, 2019

This paper investigates an optimal K-policy for a two-server Markovian queueing system M∕(M 1 , M 2)∕2∕(B 1 , B 2), with one fast server S 1 and one slow server S 2 , using the matrix analytic method. Two buffers B 1 and B 2 are organized to form waiting lines of customers in which, buffer B 1 is of finite size K(< ∞) and buffer B 2 is of infinite capacity. Buffer B 1 stalls customers who arrive when the system size (queue + service) is less than (K + 1) and dispatches a customer to the fast server S 1 only after S 1 completes its previous service. This K-policy is of threshold type which deals with controlling of informed customers and hence the customers have better choice of choosing the fast server routing through the buffer B 1. The (K + 2)-nd customer who arrives when the number of customers present in the system is exactly (K + 1) has the Hobson's choice of getting service from the slow server S 2. Buffer B 2 accommodates other customers who arrive when the number of customers present in the system is (K + 2) or more and feeds them one after another to either buffer B 1 or the sever S 2 whichever event can first accept the customer at the head-of-the-line in B 2. Queue length processes of interest are (1) q 1 = lim t→∞ X 1 (t) and (2) q 2 = lim t→∞ X 2 (t) , where X 1 (t) represents the number of customers who are in the buffers B 1 and B 2 and also in the service with server S 1 at time 't' and X 2 (t) represents the number of customers available with server S 2 only. The bi-variate random sequence (t) = (X 1 (t), X 2 (t)) of the system size (queue + service) forms a quasi-birth and death process (QBD). Steady state characteristics, and some of the performance measures such as the expected queue length, the probability that each server is busy etc are obtained. Numerical illustrations are provided based on the average cost function to explore the methodology of finding the best K-policy which minimizes the mean sojourn time of customers. Keywords QBD processes and M∕(M 1 ,M 2)∕2∕(B 1 ,B 2) • Fast server • Slow server • Matrix analytic method • Stationary distribution

International Journal of Science and Research (IJSR)

The ultimate objective of the analysis of queuing systems is to understand the behaviour of their underlying process so that informed and intelligent decisions can be made by the management. The application of queuing concepts is an attempt to minimize cost through minimization of inefficiency and delays in a system. Various methods of solving queuing problems have been proposed. In this study we have explored single –server Markovian queuing model with both interarrival and service times following exponential distribution with parameters and , respectively, and unlimited queue size with FIFO queuing discipline and unlimited customer population. We apply this model to catering data and estimate parameters for the same. A sensitivity analysis is the carried out to evaluate stability of the system.

IASET, 2013

We study the queueing system where every customer on arrival makes one of the possible decisions either to join the queue or to go away without taking service never to return. Assuming such a decision to be entirely governed by the queue size at the instant of customer arrival, the transient solution is obtained analytically using the iteration method for a state dependent birth-death queue in which potential customers.

2014

In this paper we present probability density function of vacation period of M/G/1 queueing process that operates under (0,k) vacation policy, wherein the server goes on the vacation when the system becomes empty and reopens for service immediately at the arrival of the k th customer. The number of lattice paths when last arrival is an arrival has also been derived. The transient analysis is based on approximating the general service time distribution by Coxian two-phase distribution and representing the corresponding queueing process as a lattice path. Finally the lattice path combinatorics is used to present the number of lattice paths.

Queueing Systems, 2005

In this paper, we consider a queue whose service speed changes according to an external environment that is governed by a Markov process. It is possible that the server changes its service speed many times while serving a customer. We derive first and second moments of the service time of customers in system using first step analysis to obtain an insight on the service process. In fact, we obtain an intriguing result in that the moments of service time actually depend on the arrival process! We also show that the mean service rate is not the reciprocal of the mean service time. Further, since it is not possible to obtain a closed form expression for the queue length distribution, we use matrix geometric methods to compute performance measures such as average queue length and waiting time. We apply the method of large deviations to obtain tail distributions of the workload in the queue using the concept of effective bandwidth. We present two applications in computer systems: 1) Web server with multi-class requests and 2) CPU with multiple processes. We illustrate the analysis and various methods discussed with the help of numerical examples for the above two applications.

Computers & Industrial Engineering, 2006

The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.

Operations Research, 2004

For a broad class of discrete-and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some of multi-server queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the stationary distributions of the numbers in queue and in system as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with BMAP (Batch Markovian Arrival Process) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

Queueing Systems, 2008

We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that is independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various "parts" of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that is either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little's law, to generate rigorous approximations of the steady-state queue-length in the case that the amount of work brought by a given arrival is of an arbitrary distribution.

International Journal of System and Software Engineering, 2017

In the present paper, an attempt has been made to study the optimal threshold policy for Markovian queueing model having additional servers along with permanent server. The incorporation of customer's balking and reneging behavior has been done. The customers arrive in Poisson fashion and their service times are exponentially distributed. The first server starts service when N (≥1) or more customers are accumulated and turns off when the system is empty. The (j+1)th (j=1, 2,...., r-1) server turns on when there are Nj+1 customers in the system and will be removed as soon as the number of customers drops to threshold level Nj. We use Laplace transform technique to derive transient probabilities and some other system characteristics such as the expected number of jobs in the system, throughput, and probability that jth (j=1,2,3,…,r) server being busy in rendering the service, etc.. The effects of system parameters on the performance characteristics have been examined by taking numerical illustrations.

Probability in the Engineering and Informational Sciences, 1992

We study a generalization of the GI/G/l queue in which the server is turned off at the end of each busy period and is reactivated only when the sum of the service times of all waiting customers exceeds a given threshold of size D. Using the concept of a “randomly selected” arriving customer, we obtain as our main result a relation that expresses the waiting-time distribution of customers in this model in terms of characteristics associated with a corresponding standard GI/G/1 queue, obtained by setting D = 0. If either the arrival process is Poisson or the service times are exponentially distributed, then this representation of the waiting-time distribution can be specialized to yield explicit, transform-free formulas; we also derive, in both of these cases, the expected customer waiting times. Our results are potentially useful, for example, for studying optimization models in which the threshold D can be controlled.

Queueing Systems, 1991

In this paper we consider the discrete-time single server queueing model with exceptional first service. For this model we cannot define the steady-state waiting-time distribution simply as the limiting distribution of the waiting times, since this limit does not always exist. Instead, we use the Cesaro limit to define the limiting waiting-time distribution. We give an exact relation between the generating functions of the steady-state waiting-time distribution and of the idle-time distribution in the case of general interarrival-time and service-time distributions. Once we have this relation, we can give more explicit results when the generating function of either the interarrival-time distribution or the service-time distribution is.xational. We also derive some results on the asymptotic behaviour of the waiting-time distribution.

For the M/G/1/m queue with the forced vacations in work after n customers served in succession probabilities of states of limiting steady-state process are found. 1. Introduction. In case of arbitraryly distributed service time and exponentially distributed interarrival time existence of limiting steady-state process for the single-server queueing system with unlimited queue is proved by a method of embedded Markov chains [ 1, p. 98 ]. For M/G/1/m queue probabilities of states of limiting steady-state process are found in [ 2, p. 235 ]. Below we study M/G/1/m queue which feature consists that after ï customers served in succession the forced vacations in work of system occur. Necessity of such pause can be caused by technological or other reasons. For example, the technical device which consistently carries out homogeneous operations, at designing can be calculated on the limited number of operations which are carried out without interruption between operations.