International Journal of Stochastic AnalysisVolume 2012 (2012), Article ID 673642, 20 pagesdoi:10.1155/2012/673642

Research Article

A Feedback Retrial Queueing System with Two Types of Batch Arrivals

R. Kalyanaraman

Department of Mathematics, Annamalai University, Annamalainagar 608002, India

Received 8 September 2012; Revised 15 November 2012; Accepted 18 November 2012

Academic Editor: Ho Lee

Copyright © 2012 R. Kalyanaraman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A retrial queueing system with two types of batch arrivals, called type I and type II customers, is considered. Type I customers and type II customers arrive in batches of variable sizes according to two different Poisson processes. Service time distributions are identical and independent and are different for both types of customers. If the arriving customers are blocked due to the server being busy, type I customers are queued in a priority queue of infinite capacity, whereas type II customers enter into a retrial group in order to seek service again after a random amount of time. A type I customer who has received service departs the system with a preassigned probability or returns to the priority queue for reservice with the complement probability. A type II call who has received service leaves the system with a preassigned probability or rejoins the retrial group with complement probability. For this model, the joint distribution of the number of customers in the priority queue and in the retrial group is obtained in a closed form. Some particular models and operating characteristics are obtained. A numerical study is also carried out.

1. Introduction

In the last three decades there has been significant contribution in the area of retrial queueing theory. For detailed survey one can see Yang and Templeton [1], Falin [2] and Choi and Chang [3]. Choi and Park [4] investigated an retrial queue with two type of customers in which the service time distribution for both types of customers are the same. Khalil et al. [5] investigated the above model at Markovian level in detail. Falin et al. [6] investigated a similar model, in which they assumed different service time distributions for both types of customers. In 1995, Choi et al. [7], studied an retrial queue with two types of customers and finite capacity. Atenica and Moreno [8] has analyzed a single server retrial queuing system with infinite buffer, Poisson arrivals, general distribution of service time, and linear retrial policy. If an arriving customer finds the server occupied, he joins a retrial group (called orbit) with probability and with complementary probability a priority queue in order to be served. After the customer is served completely, he will decide either to return to the priority queue for another service with probability or to leave the system forever with probability , where . They proved the ergodicity of the embedded Markov chain and obtained its stationary distribution function and the joint generating function of the number of customers in both groups in the steady-state regime. Kalyanaraman and Srinivasan [9] studied an retrial queue with geometric loss and with type I batch arrivals and type II single arrivals. Artalejo and Atenica [10] analyzed a single server retrial queue with batch arrivals. Atenica and Moreno [11] considered an retrial queue with general retrial time where the blocked customers either join the infinite waiting room (called priority queue) with probability or with complementary probability leave the service area and enter the retrial group (called orbit) in accordance with an FCFS discipline. They assume that only the customers at the head of the orbit are allowed to retry for service. They studied the ergodicity of the embedded Markov chain, its stationary distribution function, and the joint generating function of the number of customers in both groups in the steady-state regime. In 2005, Lee [12] considered an retrial queueing system with two types of customers and feedback and derived the joint generating function of the number of customers in two groups by using the supplementary variable method. Falin [13] considered a single server batch arrival queue with returning customers. In 2011, Thillaigovindan and Kalyanaraman [14] analyzed a feedback retrial queueing system with two types of arrivals and the type I arrival being in batches of fixed size .

In this paper, we deal with a feedback retrial queue with two types of customers, in which both types of customers arrive in batches of variable sizes. In Section 2, we describe the system with stability condition. In Section 3, we obtain the joint probability generating function for the number of customers in the priority queue and in the retrial group when server is busy as well as idle. The expressions for some particular models are deduced in Section 4. Some operating characteristics are derived in Section 5 and a numerical study is carried out in Section 6.

2. The Model

A retrial queueing system with two types of customers is considered in this paper. Type I customers arrive in batches of size with probability and type II customers arrive in batches of size with probability according to two independent Poisson processes with rates and , respectively. If type II customers, upon arrival, find the server busy, they enter in to an orbit of infinite capacity in order to seek service again after random amount of time. All the customers in the retrial group behave independently of each other. The retrial time is exponentially distributed with mean . Type I customers are queued in a priority queue of infinite capacity after blocking, if the server is busy. As soon as the server is free, the customers in the priority queue are served using FCFS rule and the customers in the retrial group are served only if there are no customers in the priority queue. A type I call who has received service departs the system with probability or returns to the priority queue for reservice with probability . A type II call who has received service leaves the system with probability or rejoins the retrial group with probability .

The service time distributions for both type of customers are identically and independently distributed random variables and have different distributions. A supplementary variable technique is used for the analysis and the variable considered being the residual service time of a customer in service. The service time density function is ; and , is the Laplace transformation of the distribution function .

The Stochastic process related to the model is where = number of customers in the priority queue at time , = number of customers in the retrial group at time , and = the server state at time where = the residual service time of a type customer in service at time . is a Markov process with state space and the corresponding stationary process is .

The related probabilities are

In steady state, the corresponding probabilities are and the Laplace transformation of is

It is clear that is the steady-state probability that there are customers in the priority queue, customers in the retrial group, and the server is busy with a th-type customer.

For , the following probability generating functions are defined for the analysis:

3. The Analysis

Using the mean drift argument of Falin [15], it can be shown that the system is stable if where ,.

Now the mathematical equations that govern the system are obtained by employing the remaining service time (vacation time) as the supplementary variable. Relating the state of the system at time and , the following partial differential-difference equations are obtained.

For In steady state, (3.1) becomes and the normalization condition is where .

By taking the Laplace Stieltjes transformation of (3.3) to (3.6), the following equations are obtained:

In (3.2) and (3.8), multiplying by and then summing over , the following equations are obtained:

Multiplying (3.11) and (3.13) by and summing over and using (3.10) and (3.12) leads to

By substituting in (3.14) and (3.15), we get where .

Using (3.16) and (3.17) in (3.9) and on simplification, one can get the following equation:

Define for each fixed , . By Rouche’s theorem, there is a unique solution of the equation . Now (3.18) becomes where is the root of the equation and .

Using (3.19) in (3.18), it can be seen that where , .

Using (3.20) in (3.16) leads to

Using (3.19) in (3.17) leads to

The general solution of the differential equation (3.19) is where is a constant, which is the probability that the server is idle.

Putting in (3.10) and in (3.11), we get

Summing (3.25) over to and then adding (3.24), we get

Putting in (3.12) and in (3.13) results in

Summing (3.28) over to and then adding (3.27), we get

Adding (3.28) and (3.29) and using (3.9) leads to

Evaluating at and using normalization condition we get

Using (3.31) in (3.23) gives

In steady state, the probability generating function of number of customers in the orbit when the server is idle is obtained from (3.32) and (3.23).

Substituting in (3.14)

Equation (3.33) together with (3.20) and (3.16) yields the joint probability generating function of the number of customers in the priority queue and in the orbit when the server is busy with type I customer as

Putting in (3.15),

Substituting (3.19) in (3.35), we get

Equation (3.36) together with (3.19) and (3.17) yields the joint probability generating function of the number of customers in the priority queue and in the orbit when the server is busy with type II customer as

Thus, we have the following theorem.

Theorem 3.1. The stationary distribution of has the following generating functions:

Corollary 3.2. The probability that the server busy is

4. Particular Models

By taking particular values to some parameters of the above model, the following models can be obtained.(i)When , , , and , the system coincides with that of Choi and Park [4].(ii)When , , and , the above results coincide with the results of Falin et al. [6].(iii)When , , the system coincides with that of Thillaigovindan and Kalyanaraman [14].

5. Operating Characteristics

The operating characteristics like the mean number of customers in the priority queue and the mean number of customers in the orbit have been calculated using the formulas and . After putting in (3.34) and in (3.37), we get where , , , , .

Differentiating (5.1) with respect to and then taking the limit as , we get where

After putting in (3.34) and in (3.37), we get

Differentiating (5.4) with respect to and then taking limit as , we get

From (3.31) and (3.32) (i)Mean number of customers in the priority queue is (ii)Mean number of customers in the orbit is (iii)Mean busy period.

Busy period is the length of the time interval that keeps the server busy continuously and this continues till the instant server becomes free again and let be the length of the idle period. For this model, and generates an alternating renewal process and therefore

But

Using (3.39) on (5.10), we get

6. Numerical Study

In this section, some numerical examples related to the model analyzed in this paper are given. By varying type I arrival rate, type II arrival rate, and the retrial rate, the mean number of customers in the priority queue, the mean number of customers in the orbit, the mean busy period, the probability that the server is idle, and the probability that the server is busy are calculated. For the analysis, the parameters , , , , , , , , and are fixed. In Figures 1 and 2, the retrial rate is taken as 0.9 and type II arrival rate is taken as 0.8 and 0.9, respectively; by varying values of type I arrival rate the graphs of the mean number of customers in the priority queue and the mean number of customers in the orbit are drawn. In Figures 3 and 4, the retrial rate has been fixed as 0.9 but type I arrival rates are 0.8 and 0.9, respectively. The graphs of the mean number of customers in the priority queue and the mean number of customers in the orbit are drawn against varying values of type II arrival rate. In Figures 5 and 6, the same graphs with respect to varying retrial rate are drawn for fixed values of type I and type II arrival rates (, and 0.9). Figures 7, 8, and 9 show the graphs of mean busy period for varying values of type I arrival rate and type II arrival rate and retrial rate, respectively. From Figures 7 and 8, it is seen that the mean busy period is an decreasing function with respect to type I arrival rate and the mean busy period is an increasing function with respect to type II arrival rate. Whereas from Figure 9, the mean busy period is a constant function with respect to retrial rate. In Tables 1 and 2, the probability that the server is idle and the probability that the server is busy are presented. From the graphs, it is clear that as type I arrival rate (type II arrival rate) increases the mean number of customers in the priority queue and the mean number of customers in the orbit also increase, whereas as the retrial rate increases the mean number of customers in the orbit decreases and the mean number of customers in the priority queue is a constant function.

Table 1: The probabilities and .

,  
,  

0.1
0.4511
0.0089
0.3837
0.0088

0.2
0.4422
0.0178
0.3750
0.0175

0.3
0.4334
0.0266
0.3663
0.0262

0.4
0.4246
0.0354
0.3576
0.0349

0.5
0.4158
0.0442
0.3490
0.0435

0.6
0.4071
0.0529
0.3404
0.0521

0.7
0.3984
0.0616
0.3319
0.0606

0.8
0.3898
0.0702
0.3233
0.0692

0.9
0.3811
0.0789
0.3148
0.0777

1.0
0.3725
0.0875
0.3064
0.0861

Table 2: The probabilities and .

,  
,  

0.1
0.8548
0.0777
0.8453
0.0872

0.2
0.7884
0.0766
0.7790
0.0860

0.3
0.7219
0.0756
0.7127
0.0848

0.4
0.6555
0.0745
0.6464
0.0836

0.5
0.5891
0.0734
0.5801
0.0824

0.6
0.5226
0.0724
0.5138
0.0812

0.7
0.4562
0.0713
0.4474
0.0801

0.8
0.3898
0.0702
0.3811
0.0789

0.9
0.3233
0.0692
0.3148
0.0777

1.0
0.2569
0.0681
0.2485
0.0765

Figure 1: Type I arrival rate versus mean number of customers.

Figure 2: Type I arrival rate versus mean number of customers.

Figure 3: Type II arrival rate versus mean number of customers.

Figure 4: Type II arrival rate versus mean number of customers.

Figure 5: Retrial rate versus mean number of customers.

Figure 6: Retrial rate versus mean number of customers.

Figure 7: Type I arrival rate versus mean busy period.

Figure 8: Type I arrival rate versus mean busy period.

Figure 9: Retrial rate versus mean busy period.

7. Conclusion

In the foregoing analysis, an queue with retrial queueing system with two types of batch arrivals is considered to obtain queue length distribution and mean queue length. Extensive numerical work has been carried out to observe the trends of the operating characters of the system.

Acknowledgments

The author is thankful to the referees and the editor for their valuable comments and suggestions which have helped in improving the quality of this paper.

References

T. Yang and J. G. C. Templeton, “A survey on retrial queues,” Queueing Systems, vol. 2, no. 3, pp. 201–233, 1987.

G. I. Falin, “A survey of retrial queues,” Queueing Systems, vol. 7, no. 2, pp. 127–167, 1990.

B. D. Choi and Y. Chang, “Single server retrial queues with priority calls,” Mathematical and Computer Modelling, vol. 30, no. 3-4, pp. 7–32, 1999.

B. D. Choi and K. K. Park, “The M/G/1 retrial queue with bernoulli schedule,” Queueing Systems, vol. 7, no. 2, pp. 219–227, 1990.

Z. Khalil, G. I. Falin, and T. Yang, “Some analytical results for congestion in subscriber line modules,” Queueing Systems, vol. 10, no. 4, pp. 381–402, 1992.

G. I. Falin, J. R. Artalejo, and M. Martin, “On the single server retrial queue with priority customers,” Queueing Systems, vol. 14, no. 3-4, pp. 439–455, 1993.

B. D. Choi, K. B. Choi, and Y. W. Lee, “M/G/1 retrial queueing systems with two types of calls and finite capacity,” Queueing Systems, vol. 19, no. 1-2, pp. 215–229, 1995.

I. Atencia and P. Moreno, “A queueing system with linear repeated attempts, Bernoulli schedule and feedback,” Sociedad de Estadistica e Investigacion Operativa, vol. 11, pp. 285–310, 2003.

R. Kalyanaraman and B. Srinivasan, “A retrial queueing system with two types of calls and geometric loss,” International Journal of Information and Management Sciences, vol. 15, no. 4, pp. 75–88, 2004.

J. R. Artalejo and I. Atencia, “On the single server retrial queue with batch arrivals,” Sankhya, vol. 66, pp. 140–158, 2004.

I. Atencia and P. Moreno, “A single-server retrial queue with general retrial times and Bernoulli schedule,” Applied Mathematics and Computation, vol. 162, no. 2, pp. 855–880, 2005.

Y. W. Lee, “The M/G/1 feedback retrial with two types of customers,” Bulletin of the Korean Mathematical Society, vol. 42, no. 4, pp. 875–887, 2005.

G. I. Falin, “A single-server batch arrival queue with returning customers,” European Journal of Operational Research, vol. 201, no. 3, pp. 786–790, 2010.

N. Thillaigovindan and R. Kalyanaraman, “A Feedback retrial queueing system with two types of arrivals,” in Proceeding of the 6th International Conference on Queueing Theory and Network Applications (QTNA '11), B. Kim, H. W. Lee, G. U. Hwang, and Y. W. Shin, Eds., pp. 177–181, Seoul, Korea, 2011.

G. I. Falin, “On sufficient condition for ergodicity of multichannel queueing systems with repeat calls,” Advances in Applied Probability, vol. 16, pp. 447–448, 1984.