A Study of Stochastic Model with Queue Network

INTRODUCTION

Human history reveals that great inventions have led us much of the way in which we live our lives. Novel ways have always been built to satisfy all coming demands, whether it's the invention of the wheel or the machine or IT. The use of these instruments of revolutionization has deeply affected human culture. The hunger for information and the enhancement of life quality have led us to understand the world in which we live better. Man has colonised a venture for many reasons, particularly patience of waiting for others, which is unsuccessful if he had not learnt tolerance. One of them paused and waited somewhere to join others. It was this tolerance and patience to wait for others that gave rise to the colonies. With the rise of the population of the planet, people's settlements became urban cities of different sizes. Had man been pursuing an ideology of 'immediate service on demand,' it would have led his efforts to be used unéconomically, the unity would never have formed between individual citizens, and his problems would have been numerous. He started to lead an organised life as he needed to rely on others' services to a large extent to meet his own demands. Waiting thus became a living part and waiting lines is typically made up of units, customers and messages which wait for service while it becomes obvious and predictable to have waiting lines. Gordan and Newell[1967] consider a network of Markov queues where a fixed and limited number of customers say (K) circulates across the network. There is no outside feedback or departure from the network, unlike Jackson's open network. Considering a closely connected network of "k" nodes, the contribution of the I nodes will be at the next node i+1 (1 alternatives to i-1), and the last node "k" input will be from the I node and so forth. A cyclic queue such as this is called. A cyclic queue structure consists of several series of service channels, with a closed circle at the centre. T.P. Singh Singh. [1994] broadened the research carried out by various cyclic queue network proportional to the queue numbers and heterogeneous feedback channels. This study is another widespread work done by previous researchers in the field of feedback.

This segment explores a network of three queues in sequence, input from the third server, both arriving and departing poisonously. We assume that the service rate is proportional to the amount of your queue. The equation of differential differences was tested transiently. The mean queue size and other parameters were obtained by means of statistical tools for the queue model. The model is implemented in the banking service structure, administrative installation and decision-making. The transient behaviour of the system was examined and the differential equation, thus developed in the model, solved by using the generating function technique and Lagrange's partial differential equation solution process. The aim of this section is to discuss different queue characteristics such as medium length of the queue system, variance of it, etc., and useful in the decision-making, produce, banking service, etc.

Define the probability at time (t), there are n1, n2, n3 items in system in front of nodes S1, S2, S3 (n1 n2, n3>0),

Connecting the various state probabilities at time time dependent differential difference equations for the model are obtained as:

Let the mean queue size be denoted by L. Then L as per statistical formula is defined as

The steady state solution of the model can be found in the usual manner by letting .

Also the mean queue size L of the system in the steady state case is obtained by letting in which is as

(12)

On letting and no feedback from third server, then one can easily obtain the result of the model that corresponds to the M|M|1|∞ queueing model discussed by Gross and Harries. For this if we identify Then the result (3.1.37) corresponds to On the similar line one can also easily identify the study state results corresponds as There will be no length of queue and the balance between arrival and operation pattern will be preserved. Another parameter can be determined using normal statistical formules, such as variances, distribution of busy times etc.

This segment discusses the constant status conduct of a feedback queue network in which the consumer has two options to move to the next or to obtain feedback on a previous channel after being served on a serial channel. Arrivals and departures obey each channel's Poisson law. A solution technique was introduced to define various queue parameters by generating function (i.e. f. t), L-hospital rule, laws on calculus, and statistical formulas for marginal model.

There are three service channels in accordance with the model such that one unit or customer exits the unit or joins the next channel for further activity after service at the initial channel. The products found faulty are returned to the third channel after the service has been obtained on the second service channel in a similar way. The system's constant state behaviour was investigated with the application of the generation function technique, L-hose rules and statistical formulae for the distribution of the marginal likelihood, to find different queue parameters. Poisson is believed to be distributed on arrival and departure. For a decision-making phase, the medium queue size for each channel was achieved.

The Poisson units/customers arrive at S1 in a medium μ, with a three-phase demand. The first service is completed in phase 1, and in phase 2 in phase 2 in phase 2 and in phase 3 in serial order in phase 3 in phase 3. After the first step, the units either exit S1 with probability p1 or pass through second channel S2 with probability P12 so that the units either depart with probability p2 after the service S2 or pass through the second channel p23 or feedback on the previous channel p21 with probability (where items are detected defective), so that p2+p21+p23=1. Similar feedback is obtained. The service parameters are assumed to be μ1, μ2 , μ3.

Define There are possibly n1 in the system before the S1 units, n2 units before the S2, and n3 units before the S2 units. of The standard argument leads to the following differential difference equations in steady state –

Queuing Networks is the mathematical approach for the analysis of waiting lines. A trade-off decision is central to any Queueing Network: the manager must weigh up the additional costs of offering a faster service against the inherent costs of waiting. This research introduces a new fluffy approach to Queueing systems in the event of inconsistent knowledge available to remove the drawbacks of the point evaluation and the related paradoxes when measuring the efficiency of a queue. Both the arrival rate of Poisson and the exponential service times are taken into account with fuzzy estimators. To address the main estimation problem in uncertainty, the implementation of versatile estimators in M/M/S Queueing device performance measures is presented. Queuing delay is a common phenomenon and one of the most important fields in the management of operations. It comes when the demand in a system exceeds its ability or its capacity to provide service. The first applications for queuing theory began in the telecommunications field at the beginning of the 20th century. Later several researchers generalised it and today there are various applications in daily life. Traditional problems with the Queueing Network suggest that the rate of arrival at queue and the service delivery time are dependent on Poisson and exponential probability distribution. However, if we are forced to substitute appropriate way to solve confusion in these issues. Our approach to queue efficiency avoids uncertainty prediction issues.

1. Erlang, A.K. [1909]. ―The theory of probability and telephone congestion‖ Nyt Tidsskrift for mathematic. Ser.B, Vol 20, pp. 33-39. 2. Pollaczek, F. [1930] "Uber eine Aufgabe der Wahrscheinlichkeitstheorie". Mathematische Zeitschrift, Vol. 32: pp. 64–100. 3. Crommeline, C.D. [1932] ―Delay probability formulae when the holding time is constant‖, P.O. Elec. Engg. J. Vol. 25, pp. 41-50. 4. Khintchine, A.Y. [1932] "Mathematical theory of a stationary queue". Matematicheskii Sbornik Vol.39 (4), pp. 73–84. 5. Pollaczek Felix [1934]. Concerning an Analytic Method for the Treatment of Queueing Problems [M]. Chapel Hill: North Carolina Press, 6. Brockmeyer, E., H.L. Halstrom and A. Jensen[1948]. The life and works of A.K. Erlang, the Copenhagen Telephone Company, Copenhegon, Denmark. 7. Kendall, D.G. [1951]. ―Some problem in theory of queues‖, J. Roy. Statist. Soc. Series B, Vol.13, pp. 151-185. 8. Kendall D.G, [1953]. ―Stochastic process occurring in the theory of queues and their analysis by the method of imbedded Markov Chain‖, Ann. Maths Statist Vol. 24, pp.338-354. 9. Jackson R.R.P. [1954]. ―Queuing systems with phase –type service‖, Operation Research Quart. Vol 5(4), pp. 109-120. 10. Bailey, N.T.J. [1954]. ―On Queuing process with bulk service‖, J. Roy. Statist. Soc.B, Vol. 16, pp.80-87. 11. O‘Brien, G.G, [1954] ―The solution of some queuing problems‖, J.Soc.Ind., App. Math, Vol.2, pp.133-142.

13. Hunt G. C., [1955] Sequential arrays of waiting lines. Operation Research. Vol. 4(6), pp.674-683. 14. Burke, P.J. [1956] ―The output of a queuing system‖, Oper, Ress.Vol.4, pp. 699-714.

Research Scholar, Department of Mathematics, BMU, Rohtak mamtanagwan@gmail.com