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Transient and Numerical Solution of a Feedback Queueing System with Correlated Departures

Neelam Singla*, P.C. Garg

Department of Statistics, Punjabi University, Patiala, India

Abstract: This paper studies a feedback queueing system with correlated departures. Departures take place only at transition marks. Inter-arrival times and inter-transition times follow exponential distributions. Transient-state queue length probabilities and laplace transform of the generating function of transient-state queue length probabilities are obtained. A few special cases of interest are also derived. Various probabilities relating the model are obtained numerically and are compared graphically.

Keywords: queueing, feedback, server, probability, generating function, numerical solution

Received December 12, 2013; Revised January 10, 2014; Accepted February 07, 2014

Citation: Neelam Singla, P.C. Garg, Transient and Numerical Solution of a Feedback Queueing System with Correlated Departures. American Journal of Numerical Analysis, 2 (1). 20-28. January 2014. doi:10.12691/ajna-2-1-5

1. Introduction

It was [5] who introduced the idea, in discrete time, of dependence of probability of an arrival or of no arrival at a time mark is dependent on the probability of an arrival or of no arrival at the previous time mark. Afterwards a good many research problems are solved in discrete time by [1,3,7,9,10]. Again [6] handled the same concept in continuous time. Later on [12] studied a number of random memory queueing models for the exact number of arrivals and departures by given time. These type of queues in discrete and continuous time are named in literature as correlated queues.

References [4,8] solved a number of queueing problems with feedback. However the models solved by them do not assume the dependence of probability of the happening of an event (arrival or departure) at a time mark on the previous time mark. But situations do occur in day-to-day life wherein such dependence is observed. In the present paper, a continuous time single server feedback queueing system in which probability of a departure or of no departure at a time mark is dependent on the probability of a departure or of no departure at the previous time mark, is considered.

Feedback concept is used in the sense that units are given another chance of service if required. However the units will have to leave the system definitely after getting second service. Recently [13] obtained the steady state solution of a feedback queueing system with correlated departures.

The practical situation which corresponds to the above situation can be that of a gymnasium room with some kind of exercising machine, wherein people come for exercising. It is supposed that the departures can occur only at transition points. There may arise two types of situations:

(I) Departure has occurred at previous transition mark.

Now there may be two cases according as

(a) departure occurs.

(b) departure does not occur at the present transition mark.

(II)Departure has not occurred at previous transition mark.

Now there may be two cases according as

(a) departure occurs.

(b) departure does not occur at the present transition mark.

The queueing system investigated in this paper is described by the following assumptions:

1. Arrivals are Poisson with parameter .

2. All departures take place only at the transition marks where , are identically distributed random variables.

3. If at any instance, the queue length is then the departure probabilities at two successive transition marks are governed by the following assumptions:

(a) If departure has occurred at previous transition mark then the probability of occurring a departure at present transition mark is and the probability of not occurring a departure at present transition mark is .

(b) If departure has not occurred at previous transition mark then the probability of occurring a departure at present transition mark is and the probability of not occurring a departure at present transition mark is .

4. The inter-transition times are exponentially distributed with mean transition time .

5. Units are served in their order.

6. The probability of rejoining the system is and that of leaving the system is for the units getting service first time, so that . However the units will have to leave the system definitely after getting second service.

7. The probability that the unit departs the service channel for the first time is assumed to be and that for the second time is , so that .

8. The stochastic processes involved, viz.

(a) arrival of units

(b) departure of units

(c) transition times

are statistically independent of each other.

2. Description

Probability that there are n units in the system at time t and the next unit is to depart for the first time or second time according as or and departure has occurred at the previous transition mark .

Probability that there are n units in the system at time t and the next unit is to depart for the first time or second time according as or .

= Probability that there are units in the system at time and departure has occurred at the previous transition mark

= Probability that there are units in the system at time .

Initially,

Elementary probability reasoning leads to the following difference differential equations

[picture] | (1)

[picture] | (2)

[picture] | (3)

[picture] | (4)

[picture] | (5)

[picture] | (6)

Taking the Laplace Transformation ; Re s > 0 of (1)-(6).

[picture] | (7)

[picture] | (8)

[picture] | (9)

[picture] | (10)

[picture] | (11)

[picture] | (12)

3. Mathematical Analysis of the Problem

Description of the generating functions

and

all for or with

Laplace Transformation of Probability Generating Function of transient-state queue length probabilities

[picture] | (13)

[picture] | (14)

[picture] | (15)

[picture] | (16)

[picture] | (17)

Where

Let

where

Obviously have at least one zero inside the unit circle.

Now

for

Hence on

Since all the conditions of Rouche’s theorem are satisfied, so has at least one zero inside the unit circle. Let it be . Numerator must also vanish for this zero since , and are analytical functions of . From this we will get one equation in six unknowns and . Eliminating common factor from equations (10) and (12), we get second equation in above six unknowns for . Other four equations required for evaluating six unknowns are equations (8) & (11) and equations (7) & (9) for .

Hence the generating functions , and are completely known. and can be obtained by using the following formulae

at for

at for

at for

at for

at for

In either case , and can be found by inverting the Laplace transforms , and using [2].

Further , as desired

And

Also & .

4. Special Cases

Case (i). When the units are not given the option of rejoining the system after first service i.e. a queueing system without feedback when departures at two successive transition marks are correlated with inter-arrival times and inter-transition times following exponential distribution. Putting , ; , , , , , ; in (13), (15, and (17) and on simplification , we get

[picture] | (18)

[picture] | (19)

[picture] | (20)

Case (ii). When departures at two successive transition marks are independent i.e. , then (13) to (17) give

[picture] | (21)

[picture] | (22)

[picture] | (23)

[picture] | (24)

[picture] | (25)

5. Numerical Solution and Graphical Representation

Following the work of [11], the numerical results are generated from equations (1)–(6) by Runga Kutta Method of order four for solving ordinary differential equations using MATLAB programming. In Figure 1 to Figure 8, various probabilities are plotted versus time for the case when .

Figure 1.

Figure 2.

Figure 3.

Figure 2 shows relative change in probabilities , , and . All these probabilities first increase then decrease to some extent and finally attain some steady values for higher values of . Figure 3 and Figure 4 show respectively the probability plots of , , , and , , , versus time. All these probabilities firstly increase and finally settle down to some steady values.

Figure 4.

Figure 5.

Figure 6.

Comparison among four probabilities when no departure has occurred at the previous transition mark and the unit at the head of the queue is to join the server for the first time i.e. among , , and is done through Figure 5. It is apparent that probability decreases as increases. It is also seen that all the four probabilities increase rapidly in the starting moments, then decreases to some extent and all show slight variability before finally approaching to some steady values. Figure 6 shows similar type of relationship among probabilities , , and as shown by Figure 5 among , , and with a difference that decrease in is not that much pronounced as is there in case of (Figure 5). It is also observed that as increases, probabilities increase with less rapidity in their initial times as compared to the probabilities with lower values.

Figure 7 illustrates the relationship among four probabilities when a departure has occurred at the previous transition mark and the unit at the head of the queue is to join the server for the first time i.e. among , , and . It shows that probability decreases as (number of units in the system) increases. It is also seen that increases rapidly in the starting moments, then decreases to some extent for higher values of (average transition times) and show slight variability before finally approaching to a steady value. Other three probabilities increase rapidly in the initial moments, then increase slowly and finally attain some constant values for higher values of . Figure 8 shows similar type of relationship among probabilities , , and when plotted against time as is shown by the Figure 7 among , , and .

Figure 7.

Figure 8.

To study the effect of various parameters on different probabilities of the model, data of various probabilities is generated for different values of keeping other parameters constant. The set of values that took is {0.2, 0.3, 0.5, 0.8}. The other parameters were fixed at . The probability is plotted against time for different values of in Figure 9. From the figure it is concluded that the probability decreases with increasing . So more the i.e. more customers are arriving per unit average transition time less is the probability of zero units in the system.

Figure 9.

Figure 10.

Figure 11.

Behavior of the probabilities and , with respect to time (average transition times) and changing is apparent from Figure 10 to Figure 11.

Effect of change in probability (probability of leaving the system after getting first service) is studied by generating data of various probabilities for different values of probability keeping other parameters constant. The set of values that took is {0.45, 0.6, 0.75}. The other parameters were fixed at . In Figure 12, the probability is plotted against time for different values of . From the figure it is clear that there is no significant change in the value of probability as increases. So no significant effect of varying probability is observed when there is no unit in the system.

Figure 12.

Figure 13.

Figure 13 shows probability plot of with respect to (average transition times) for a set of values. It is clear from the graph that in all the three cases probability increases rapidly in the starting moments then decreases and finally becomes almost steady for higher values of . It is observed that probability takes higher values for high value of . This can be explained as (i.e. zero state) is more probable when value is high. Probability is plotted vs time in Figure 14. Here it is observed that probability takes smaller values for high value of as (i.e. one state) is less probable when takes a high value.

Figure 14.

Figure 15 & Figure 16 show the probability plots of and . These also show similar kind of relationship as shown by probabilities and in Figure 13 & Figure 14 respectively and also can be explained by same argument.

Figure 15.

Figure 16.

From the above discussion and on observing all the above graphs, it is interpreted that all the probabilities finally tend to attain some steady value for high values of time (average transition times). It is also checked numerically that at any moment sum of all probabilities related to the queueing system is one.

References

[1]   | N.N. Agarwal, “Some Problems in the Theory of Reliability and Queues”, Ph.D. Thesis, Kurukshetra University, Kurukshetra, 1965.
[2]   | Bateman, H. Tables of Integral Transforms, Vol. 1, McGraw-Hill Book Company, New York, 1954.
[3]   | M.L. Chaudhry, “Some Queueing Problems with Phase Type Service, Operations Research”, Vol. 14, No. 3, 1966.
[4]   | P.C. Garg, “A Measure to Some Time Dependent Queuing Systems without/with Feedback”, Ph.D. Thesis; Kurukshetra University, Kurukshetra, 1988.
[5]   | C. Mohan, “Some problems in the Statistical theory of Random Walks”, Ph.D. Thesis, London School of Economics, U.K, 1958.
[6]   | Mohan and Murari, K., “Time Dependent Solution of a Correlated queueing problem with variable capacity”, Metrika, Vol. 19, pp 209-215, 1972.
[7]   | Murari, K., “A Queueing Problem with arrivals in batches of variable size and service rate depending on queue length”, ZAAM, 49, pp. 157-162, 1969.
[8]   | Sharda and Garg, P.C., “An M/M/1/∞ Queueing System with Feedback”, Microelectron. Reliability 26, 261-264, 1986.
[9]   | Sharda and Rana, “A queueing problem with random memory arrivals and heterogeneous servers”, Microelectron Reliability 25, 645-650, 1985.
[10]   | Tuteja, R.K., “A Queueing Problem with arrivals correlated and finite number of servers”, CORS, Vol. 4, No. 3, 1966.
[11]   | Bunday, B.D., Basic Queueing Theory, Edward Arnold (Publishers) Ltd., London, 1986.
[12]   | Garg, I, “An Approach to Some Continuous Time Queueing Problems”, Ph.D. Thesis, Kurukshetra University, Kurukshetra, 1985.
[13]   | Garg, P.C. and Singla, Neelam, “A Feedback Queueing System with Correlated Departures”, Int. J. Agricult. Stat. Sci, Vol. 8, No. 2, 2012.