Quasi-birth and death processes of two-server queues with stalling (original) (raw)
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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.
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.
Analysis of Three Server Queues with Stalling
— This article analyses a Markovianqueueing system M/(M 1 ,M 2 ,M 3)/3/(B 1 ,B 2) with stalling. It stalls customers ofqueue-1 into a finite buffer B 1 of maximum size 'K<∞' and accommodates all other waiting customers ofqueue-2 into an infinite buffer B 2. There are three heterogeneous servers labelled as 'S 1 , S 2 and S 3 ' with exponential rates 1 , 2 , and 3 respectively where 1 >> 2 > 3. Arrivals occur according to a Poisson process with mean arrival rate. Main focus is on the steady state queue length distribution which consists of two stages. Instage-1, 'Queue Length' process of the finite M/(M 1 ,M 2 ,M 3)/3/(B 1 ,3) system is formulated as a Quasi-Birth and Death process in a three dimensional finite state space. The stationary probability vector of the queue length is obtained using matrix analyticalmethods. In stage-2, using analytical methods, stage-1 results obtained on the finite queue length are linked to the infinite queue length process of the M/(M 1 ,M 2 ,M 3)/3/(B 1 ,B 2) system subject to a condition ρ=((//)<1 where = 1 + 2 ++ 3. Further steady state expressions are found for some of the performance measures such as the expected queue length, the probability that each server is busy etc. Results for some special systems have been obtained from these computational methods. A numerical study is then carried out to support the advantages of the proposed methodology.
Queueing analysis and optimal control of and systems
Computers & Industrial Engineering, 2009
We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process ðBMAPÞ: The server serves customers in batches of maximum size 'b' with a minimum threshold size 'a'. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N Ã of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process ðMSPÞ: Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.
M/Ck/1 Queue with Impatient Customers
Stochastic Processes and Models in Operations Research, 2000
A single-server queuing system with impatient customers and Coxian service is examined. It is assumed that arrivals are Poisson with a constant rate. When the server is busy upon an arrival, customer joins the queue and there is an infinite capacity of the queue. Since the variance of the service time is relatively high, the service time distribution is characterized by k-phase Cox distribution. Due to the high variability of service times and since some of the services take extremely long time, customers not only in the queue, but also in the service may become impatient. Each customer, upon arrival, activates an individual timer and starts his patience time. The patience time for each customer is a random variable which has exponential distribution. If the service does not completed before the customer's time expires, the customer abandons the queue never to return. The model is expressed as birth-and-death process and the balance equations are provided.
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.
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.
Journal of Industrial & Management Optimization, 2017
This article studies an infinite buffer single server queueing system under \begin{document}$ N enddocument−policyinwhichcustomersarriveaccordingtoadiscrete−timebatchMarkovianarrivalprocess.Theservicetimesofcustomersareindependentandobeyacommongeneraldiscretedistribution.Theidleserverbeginstoservethecustomersassoonasthequeuesizebecomesatleastbegindocument\end{document}-policy in which customers arrive according to a discrete-time batch Markovian arrival process. The service times of customers are independent and obey a common general discrete distribution. The idle server begins to serve the customers as soon as the queue size becomes at least \begin{document}enddocument−policyinwhichcustomersarriveaccordingtoadiscrete−timebatchMarkovianarrivalprocess.Theservicetimesofcustomersareindependentandobeyacommongeneraldiscretedistribution.Theidleserverbeginstoservethecustomersassoonasthequeuesizebecomesatleastbegindocument N $\end{document} and serves the customers until the system becomes empty. We determine the queue length distribution at post-departure epoch with the help of roots of the associated characteristic equation of the vector probability generating function. Using the supplementary variable technique, we develop the system of vector difference equations to derive the queue length distribution at random epoch. An analytically simple and computationally efficient approach is also presented to compute the waiting time distribution in the queue of a randomly selected customer of an arrival...
Probability Density Function of M/G/1 Queues Under (0,K) Control Policies: A Special Case
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.