Analysis of Three Server Queues with Stalling (original) (raw)
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Quasi-birth and death processes of two-server queues with stalling
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
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.
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.
ON THE DISTRIBUTION OF THE NUMBER OF STRANDED CUSTOMERS IN A M/M(b,b)/1 QUEUEING SYSTEM
The paper studies a queuing model with Poisson arrival process and bulk service. The server serves the customers in batches of fixed size b, and the service time is assumed to be exponentially distribution. The model is analyzed to find the steady-state distribution of the number of customers stranded following each service. The approach adopted is based on discrete-time Markov chains, instead of Laplace transforms that is usually used in literature. A simulation study is carried out to estimate the expected number of stranded customers at any point of time, its variance and the downside risk for given values of the system parameters.
2016
Queueing Theory is one of the most commonly used mathematical tool for the performance evaluation of systems. The aim of the book is to present the basic methods, approaches in a Markovian level for the analysis of not too complicated systems. The main purpose is to understand how models could be constructed and how to analyze them. It is intended not only for students of computer science, engineering, operation research, mathematics but also those who study at business, management and planning departments, too. It covers more than one semester and has been tested by graduate students at Debrecen University over the years. It gives a very detailed analysis of the involved queueing systems by giving density function, distribution function, generating function, Laplace-transform, respectively. Furthermore, Java-applets are provided to calculate the main performance measures immediately by using the pdf version of the book in a WWW environment. I have attempted to provide examples for ...
Queues with slow servers and impatient customers
European Journal of Operational Research, 2010
We study M=M=c queues (c ¼ 1, 1 < c < 1 and c ¼ 1Þ in a 2-phase (fast and slow) Markovian random environment, with impatient customers. The system resides in the fast phase (phase 1) an exponentially distributed random time with parameter g and the arrival and service rates are k and l, respectively. The corresponding parameters for the slow phase (phase 0) are c, k 0 , and l 0 ð6 lÞ. When in the slow phase, customers become impatient. That is, each customer, upon arrival, activates an individual timer, exponentially distributed with parameter n. If the system does not change its environment from 0 to 1 before the customer's timer expires, the customer abandons the queue never to return.
A Queueing Model of General Servers in Tandem with Finite Buffer Capacities
International journal of operations research, 2004
⎯We consider a queueing model with finite capacities. External arrivals follow a Coxian distribution. Due to the limitation of the capacity, arrivals may be lost if the buffer is full. Our goal is to study the probability of blocking. In order to obtain the steady-state probability distribution of this model, we construct an embedded Markov chain at the departure points. The solution is solved analytically and its analysis is extended to semi-Markovian representation of performance measures in queueing networks.
Performance Analysis of A Queueing System with Server Arrival and Departure
ACM SIGMETRICS Performance Evaluation Review
In many systems, in order to fulfill demand (computing or other services) that varies over time, service capacities often change accordingly. In this paper, we analyze a simple two dimensional Markov chain model of a queueing system in which multiple servers can arrive to increase service capacity, and depart if a server has been idle for too long. It is well known that multi-dimensional Markov chains are in general difficult to analyze. Our focus is on an approximation method of stationary performance of the system via the Stein method. For this purpose, innovative methods are developed to estimate the moments of the Markov chain, as well as the solution to the Poisson equation with a partial differential operator.
Systems with Queueing and their Simulation
2011
In the queueing theory, it is assumed that customer arrivals correspond to a Poisson process and service time has the exponential distribution. Using these assumptions, the behaviour of the queueing system can be described by means of Markov chains and it is possible to derive the characteristics of the system. In the paper, these theoretical approaches are presented on several types of systems and it is also shown how to compute the characteristics in a situation when these assumptions are not satisfied Keywords—Queueing theory, Poisson process, Markov chains.