A Study on Geo/G/1 Retrial Queueing System with Starting Failure and Customer Feedback

INTRODUCTION

In the past two decades, significant contributions have been made to the analysis of queueing models with recurrent customers. Readers may refer Artalejo (1999), Gomez-Corral (2006) and Artalejo (2010) for the bibliography on retrial queues. The researchers in the field of retrial queues were mainly focused on the continuous time models. But. as a result of technological growth in the computer and telecommunication systems, at the end of twentieth century researchers started to analyze retrial queues in discrete time. Bruneel and Kim (1993), Takagi (1993) and Woodward (1994) discuss the discrete time queues in detail. It is observed that the discrete time queues are more appropriate to analyze the systems that have digital basic units such as bits, packets, slots and machine cycle time. The discrete time queues have been discussed by several researchers initiated by Meisling (1958) and further studied by

Atencia and Moreno (2004a, 2006), Artalejo and Gomez-Corral (2008) and Wu et al. (2013). Nobel (2016) presented a short survey on retrial queueing models in discrete time. Atencia (2017) has studied a Geo/G/1 retrial queueing system with priority services. Upadhyaya (2018) has done a performance analysis of a Geo/G/1 retrial queue under J-vacation policy. A single-server retrial queue with server vacation was studied by Gao and Wang (2019) wherein they considered two waiting buffers to model ATM networks. Recently, Lan and Tang (2019) have analyzed the performance of a "Discrete time Geo/G/1 retrial queue with non-pre-emptive priority, working vacations and vacation interruption”. Kulshrestha and Shruti 1 (2020) analyzed a discrete time to model communication networks with second optional service and negative user arrivals. Lan and Tang (2019) have dealt with a discrete time Geo/G/1 retrial queueing system with probabilistic pre-emptive priority and balking customers, in which the server is subject to starting failures and replacements in the repair times may occur with some probability.

Server breakdown is an inevitable phenomenon in any service facility. But most of the papers in the queueing literature consider the queueing models where the servers are never subjected to failure, which is practically unrealistic. Actually, queueing systems with servers subjected to starting failures are a natural and important idea for modeling server unavailability due to external causes. Wang and Zhao (2007) studied a discrete time queue where the server may not be started successfully every time when the customer approaches for service. Customer impatience is one of the important features of any real time queueing situation, as customers do not like to spend more time on just waiting to be served - i.e., they may not join the system if they expect a long wait. In view of the importance of this balking phenomenon in the communication systems, many researchers are now paying attention to the queueing models with customer balking (Aboul-Hassan et al., 2005, 2008; Wei et al., 2015). Recently, Madheswari et al. (2019) have discussed an M/G/l retrial queue in which they have taken customer balking as one of the constraints. In our present study we take into account the customer’s dilemma whether to join the system or balk. The erroneous or lost messages are retransmitted in any communication system, and if a customer is not satisfied with the service received then the server may repeat the call in any call center facility. These situations are well studied by modelling them as a queueing system with customer feedback. Introduced by Takacs (1963), the feedback concept has been studied in continuous time (Choi and Kim, 1997; Krishna Kumar et al., 2002; Madan and Ai-Rawwash, 2005, etc.,) and also discussed for discrete time queueing systems (Atencia and Moreno, 2004b; Atencia et al., 2009).

The practical motivation for the model considered is the wireless sensor networks (WSN) deployed to monitor the forest fire or in the military surveillance that operates in discrete time environment. The WSN has a number of sensor nodes and a sink node. The sensor nodes communicate the data that it detected to the sink node through wireless links for further processing. The sensor nodes act as the server, and there is a possibility of failure while starting the transmission of the sensed data. The data packets not delivered to the sink node will be retransmitted. When a data packet arrives at a sensor node for transmission but the node is busy or down, the data may be lost. In this scenario of WSN the data to be transmitted, the sensor node, undelivered packets retransmitted, the data finding the sensor node busy is trying again and a few data are lost when the node is busy or down correspond to customer, server, feedback, retrial and balking in queueing terminology, respectively.

In this chapter, we analyze the retrial queueing system with customer feedback and balking where there is a possibility of server failure while starting the service in discrete time. We consider an early arrival system (EAS) where the departure has precedence over the arrivals. We expect this work to be the discrete time counterpart of the study on “M/G/l retrial queueing system with customer balking and feedback where the server is prone to starting failure”. The remainder of the chapter is organized as: Our model is described in detail in section 6.2. The Markov chain of our model is analyzed in section 6.3. Marginal generating functions are derived of the orbit size for different server states. The decomposition property is established in section 6.4 for the system size distribution. In section 6.5, we give graphical illustrations to show the impact of chosen parameters on some quantities of interest. Section 6.7 provides the concluding remarks.

OUR MODEL

We examine a discrete time single-server queueing system with recurrent customers in which the time line is divided into small subintervals denoted by 0,1,2.... Arrivals, departures, retrials, server failures and repair completion are assumed to occur only near the end of the subintervals. Further, all queueing activities that make the server become available (departures and repair completion) occur in (Г, t), and activities that make the server not available to serve the customers (arrivals, retrials, server failure) occur in (t, t+) in the same order. These are the characteristics of the “early arrival system EAS”. Primary arrivals from outside follow a geometrical distribution with probability a. If the server is not available for providing service, the customer chooses to join the orbit with probability b or to balk (leave the system once for all) with probability b. On the other hand, if the server is available upon arrival of a customer, his service begins at once. The customers waiting in the orbit are permitted to try again for their service as per the FCFS discipline. It is assumed that the inter retrial times follow an arbitrary distribution {r/,}"= о- The generating function

is L(x) = ) r,,xh. If the server is idle when a new or retrial customer arrives, /1=0

he should “turn on” the sever to receive his service. With probability 6 the server is turned on and starts serving the customer immediately. On the other hand, the server fails to start with probability 0 = 1-9 and the server undergoes repair. In this case, the customer who attempted to start the server has to join the orbit. The service times are assumed to be independent and identically distributed (i.i.d) random variables

bjXJ j=i

and n"' factorial moments |3L„. The customer decides to leave the system after service completion with probability v or to join the orbit for another service with probability v (v + v = 1). The repair times are also i.i.d random variables and assumed to fol-

Xoo

b2 )xJ and n"' j=i

factorial moments p2,„. We suppose 0

Throughout the chapter, we will denote by a = 1 - a. The load of the system is given by pi +p2 where pi =apu;p2 =flp2.i.

THE MARKOV CHAIN

The model proposed in the previous section is described by discrete time Markov process Y-, = (CjXojXi.iXijyNj) at time i+ where

and Nj denotes the number of retrial customers. If C, = 0, then ^o.i denotes the remaining retrial time, when C, = 1, then £i.. denotes the remaining service time and if C, = 2, then £2,; corresponds to the time required for completion of repair. The process {T(,t e N) can be shown as the Markov chain of the model under study. {(0,0);(0,/j,g):/t > l,g > l;(l,A,g): h > 1 ,g > 0;(2,/t,g): h > l,g > 1).

It is required to derive the following steady-state distribution:

of the Markov chain Yt = (CjXojXijXu^i)-

The system of Kolmogorov equations for the stationary distribution is obtained as

where normalization condition is

For solving the Eqs. (6.1)—(6.4), the PGFs are defined as:

and the auxiliary generating functions as:

Multiplying Eqs. (6.2)-(6.4) by zs and taking summation over g, we have

Substituting Eq. (6.1) in Eqs. (6.5)-(6.7), we get

Multiplying Eqs. (6.8)-(6.10) by xb and taking summation over h, we obtain Choosing x = 1 in Eq. (6.11), it reduces to

Now using the above equation in Eqs. (6.12) and (6.13), we obtain

Setting x = a in Eq. (6.11) and x = abz + ab + a in Eqs. (6.14) and (6.15), we get Using mathematical manipulations the generating functions are obtained as

where Q(z) = [(v + vz)6Bi (abz + ab + a) + Q:B2{abz + ab + a)][z + (1 - z.)aL(a)]- z(,abz + ab + a).

Lemma 1

1. The inequality [QB{(abz + ab + a)(v + vz) + BzJ}2(abz + ab + a)]

[z. + (-z)aL(a)]-z(abz + ab + a)>0 holds for 0)q + 0v + 0 < ab + aL(a).

2. The following limit exists if and only if (0p, + 0p2 )b + 0v + 0 < ab + aL(a), Proof. Define:

We find the derivatives of the function u(z) and v(z) at z = 1 to examine the slope of their tangents.

Here, it can be observed that «'(1) < v'(l). Therefore, we have u(z) > v(z) in 0 < z, < 1 since и and v are convex functions. Both the limits in statement (2) exist provided denominator is nonzero - i.e., (0pi +0p2)b+ Q Finally, the theorem given below discusses the solutions of the stationary equations.

Theorem 1

The PGFs for the steady-state distribution of the Markov chain are given by where

Proof. Substituting Eqs. (6.19)—(6.21) into Eqs. (6.11), (6.14) and (6.15) we derived the generating functions (pj(x,z),j = 0,1,2 as given in the statement of the theorem. Making use of P(0) + 0(l, 1) -I- cp,(1,1) + tp2(l, 1) = 1, we calculate P(0) as:

Making use of the above theorem, we obtain the results stated in the Corollary 1. Corollary 1

1. The marginal PGF for the orbit size given that the server is free:

2. The marginal PGF for the orbit size given that the server is serving a customer:

3. The marginal PGF for the orbit size given that the server is under repair:

4. The PGF of the orbit size:

5. The PGF for the system size:

Using Eqs. (6.24)-(6.27), we derive Eqs. (6.28)-(6.30) and from those results after simple mathematics we obtain Eq. (6.32).

Remark: The M/G/l retrial queue with feedback and starting failures discussed by Krishnakumar et at. (2002) can be approximated by our discrete time retrial queue under study. Suppose that the time is divided into intervals of equal length Д. The parameters for continuous time system can be approximated as:

Taking b = 0 (no balking), our results agree with the continuous time counterpart (see, Krishnakumar et al., 2002).

Performance Measures

Here, we present a few important quantities of interest that we derived for the system under consideration.

1. Probability that the system is free:

2. Probability that the system is occupied:

3. Probability of the server being free:

4. Probability of the server being busy:

5. Probability of the server is under repair:

6. Expected orbit size:

where

7. Expected system size:

where

8. Expected waiting time:

Deductions

We deduced a few earlier results as the special case of model proposed in this chapter. Case (i): As we let b = l,

where

The above is the PGF of the system size for the system with starting failure and feedback of customers, which coincides with [17].

Case (ii): When v = l,(p(z) reduces to

where

The above is the PGF of the system size in “the standard Geo / G /1 / ~ queue with starting failures and balking customers”.

Case (iii): When v = 1, cp(z) has the expression

This is the PGF for the system size in “the Geo / G /1 queue with impatient customers and server subjected to starting failures”.

Case (iv): When 9 = b = v = 1, our model reduces to “Geo / G /1 retrial queue with reliable server” and we have

with

which are exactly the generating functions of Theorem 1 in Atencia and Moreno (2004a).

 
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