Complete analysis of finite and infinite buffer queue—A computational approach (original) (raw)
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We first consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process () BMAP. 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. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Next we obtain queue-length distributions at various other epochs such as, pre-arrival, arbitrary and pre-service using relations with post-departure epoch. Later we also obtain the system-length distributions at post-departure and arbitrary epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue-lengths and mean waiting times have been obtained. Total expected cost function per unit time is also derived to determine the locally optimal values of a and b. Secondly, we perform similar analysis for the corresponding infinite-buffer single server queue where arrivals occur according to a BMAP and service process in this case follows a non-renewal one, namely, Markovian service process ().
Delay analysis of a queue with re-sequencing buffer and Markov environment
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There are simple service disciplines where the system time of a tagged customer depends only on the customers arrived to the system earlier (e.g. FIFO) or later (e.g. LIFO) than the tagged one. In this paper we consider single server queueing system with two infinite queues in which the system time of a tagged customer may depend on both the customers arrived to the system earlier and later than the tagged one. New regular customers arrive at the system according to MAP flow, occupy one place in buffer and receive service in FIFO order. External re-sequencing signals also arrive at the system according to (different) MAP flow. Each re-sequencing signal transforms one regular customer into delayed one by moving it to another queue (resequencing buffer), wherefrom it is served with lower priority than the regular ones. Service times of customers from both queues also have MAP distribution different from those which govern arrivals. Similar queueing system has been analysed with memoryless ingredients (arrival, service, re-sequencing). In this paper we investigate how the essential analytical properties of scalar functions, which made the analysis of the memoryless system feasible, can be extended to the case of Markov environment.
Mathematical Methods of Operations Research, 2008
In this paper, we study the behavior of a discrete-time multiserver buffer system with infinite buffer size. Packets arrive at the system according to a two-state Markovian arrival process. The service times of the packets are assumed to be constant, equal to multiple slots. The behavior of the system is analyzed by means of an analytical technique based on probability generating functions (PGF's). Explicit expressions are obtained for the PGF's of the system contents and the packet delay. From these, the mean values, the variances and the tail distributions of the system contents and the packet delay are calculated. Numerical examples are given to show the influence of various model parameters on the system behavior.
Analyzing state-dependent arrival in GI/BMSP/1/∞ queues
Mathematical and Computer Modelling, 2011
We consider an infinite-buffer single-server queue with renewal input. The service to the queueing system is provided in batches of random size, according to a batch Markovian service process (BMSP). The queue length distribution of the number of customers in the system at pre-arrival and arbitrary epochs has been obtained along with some important performance measures, such as the mean number of customers in the system and the mean system sojourn time of a customer. Secondly, we study a similar queueing system with queue-length-dependent inter-arrival times and obtain the abovementioned state probabilities and performance measures. These queueing models have potential applications in the areas of computer networks, telecommunication systems, manufacturing systems, etc.
Advances in Operations Research, 2015
We analyze an infinite-buffer batch-size-dependent batch-service queue with Poisson arrival and arbitrarily distributed service time. Using supplementary variable technique, we derive a bivariate probability generating function from which the joint distribution of queue and server content at departure epoch of a batch is extracted and presented in terms of roots of the characteristic equation. We also obtain the joint distribution of queue and server content at arbitrary epoch. Finally, the utility of analytical results is demonstrated by the inclusion of some numerical examples which also includes the investigation of multiple zeros.
Mathematics
In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under (a,b)-bulk service rule. The queueing system has a finite-buffer capacity ‘N’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distri...