PERSPECTIVES ON BALKING AND RENEGING IN M/M/1/K QUEUES

The analysis of a single server finite buffer Markovian queuing system with state-independent balking and Markovian reneging has been presented in this paper. Reneging till beginning and reneging till end of service have been considered separately. Closed form expressions of number of performance measures have been presented. Sensitivity analysis has also been performed to examine the variation in performance measures with the variation in system parameters. A numerical example has been presented to demonstrate results derived.

Similarly, an early work on balking was by Haight (1957). Haghighi et al (1986) considered a Markovian multiserver queuing model with balking as well as reneging. Each customer had a balking probability which was independent of the state of the system. R_BOS was considered as reneging discipline. Liu et al (1987) considered an infinite server Markovian queuing system with reneging of type R_BOS. Customers had a choice of individual service or batch service: batch service being preferred by the customer. Brandt et al (1998) considered a S-server system with two FCFS queues, where the arrival rates at the queues and the service may depend on number of customers 'n' being in service or in the first queue, but the service rate was assumed to be constant for n>s. The customers in the first queue were assumed impatient customers with deterministic reneging. Wang et al (1999) considered the machine repair problem in which failed machines balk with probability (1-b) and renege according to a negative exponential distribution. Another work using the concepts of balking and reneging in machine interference queue has been carried out by Al-Seedy and Al-Ibraheem (2001).
There have been a few papers in which both balking as well as reneging was considered. Here mention may be In this paper we discuss the analysis of M/M/1/k model with the additional restriction that a customer may balk as well as renege. The two types of reneging R_BOS and R_EOS are discussed separately. To the best of our knowledge, an analysis with both types of reneging (of Markovian type) in addition to state independent balking has not been carried out in this model. Even though one can find queuing models of various types analyzed in queuing literature, it is not often that reneging and balking have been analyzed together. Even if these have been dealt with, explicit closed expressions are not available. This paper is an attempt in this direction. Importance of the queuing model stems from the fact that in the classical M/M/1 model," it is assumed that the system can accommodate any number of units. In practice, this may seldom be the case. We have thus to consider the situation such that the system has limited waiting space and can hold a maximum number of k units (including the one being served)" {Medhi (1994)}. As for balking, we assume that each customer has a state independent balking probability. It will be assumed that if the customer on arrival observes the existence of a queue but with system size below finite buffer restriction, the probability that he will balk is 'p'. It may be noted that our formulation requires that balking is possible only when the system is non-empty. There is no balking from an empty system. We further assume that each customer joining the system has a random patience time following exp ( ) distribution. This patience time commences from the time it joins the systems. In case the reneging distribution is R_BOS, the customer will renege i.e. leave the system in case service does not begin before expiry of this patience time. In case of R_EOS, the customer would renege in case service is not over before the expiry of the patience time. Thus in case of R_EOS, the customer may depart either from the queue or from the service station with partial and incomplete service whereas in case of R_BOS, the customer can renege only from the queue. The arrival and service patterns are assumed to be Markovian with rates λ and μ respectively.

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The rest of the paper is structured as follows. In section 2, we obtain the system state probabilities. In section 3, performance measures are presented. Sensitivity analysis is carried out in section 4. A numerical example is discussed in section 5. Section 6 concludes the paper. Some derivations are given in the appendix which is placed in section 7.

The System State Probabilities:
In this section, the steady state probabilities are derived. Under R_BOS, let p n denote the probability that there are 'n' customers in the system. Applying the Markov process theory, we obtain the following set of steady state equations. (2.5) Under R_EOS, let q n denote the probability that there are n customers in the system. Proceeding similarly, we obtain the following set of steady state equations.

Performance Measures:
An important measure is 'L', which denotes the mean number of customers in the system. To obtain an expression for the same, we note that L=P΄(1) where Here P(S) is the p.g.f. of the steady state probabilities. The derivation of P΄(1) is given in the appendix. From (7.1.1) and (7.2.1), the mean system size under two reneging rules are Mean queue size can now be obtained and are given by Customers arrive into the system at the rate of λ. However all the customers who arrive do not join the system either because of balking or because of finite buffer restriction. The effective arrival rate into the system is thus different from the overall arrival rate and is given by where 'p k ' and 'p 0 ' are given in (2.4) and (2.5) respectively. Similarly where 'q k ' and 'q 0 'are given in (2.9) and (2.10) respectively.
It is relevant to note here that unlike in the traditional M/M/1 model where it is important that λ<μ, something similar need not hold in M/M/1/k model as customers arriving after maximum buffer size has been reached are turned back.
We have assumed that each customer has a random patience time following exp (ν). Clearly then the reneging rate of the system would depend on the state of the system as well as the reneging rule. The average reneging rate (avg rr) is a measure of average loss due to reneging. For the two rules of reneging, it is obtained as To the system manager, customers who balk or renege represent business lost. In totality, customers are lost to the system in three ways, due to finite buffer, due to reneging and due to balking. The management would like to know the proportion of total customers lost in order to have an idea of total business lost.
The mean rate at which customers are lost (under R_BOS) is Similarly the mean rate at which customers are lost (under R_EOS) is These rates help in the determination of proportion of customers lost which is of interest to the system manager as also an important measure of business lost. This proportion (under R_BOS) is given by and the proportion (under R_EOS) is given by The proportion of customers completing receipt of service can now be easily determined from the above proportion.
The customers who leave the system do not receive service. Consequently, only those customers who reach the service station constitute the actual load of the server. From the server's point of view, this provides a measure of the amount of work he has to do. We shall denote by λ s , the rate at which customers reach the service station. Then under R_BOS λ s (R_BOS) = λ e (R_BOS) (1-proportion of customers lost due to reneging out of those joining the system) In case of R_EOS, one needs to recall that customers may renege even while being served and only those customers who renege from the queue will not constitute any work for the server. Then λ s (R_EOS) = λ e (R_EOS) (1-proportion of customers lost due to reneging from thequeue out of those joining the system)

Numerical Example:
To illustrate the use of our results, we apply them to a queuing problem. We quote below an example from Allen (2005, page 267 and 273).
'Traffic to a message switching centre for Extraterrestrial Communications Corporation arrives in a random pattern (remember that 'random pattern' means exponential interarrival time) at an average rate of 240 messages per minute. The line has a transmission rate of 800 characters per second. The message length distribution (including control characters) is approximately exponential with an average length of 176 characters. Calculate the principal statistical measures of system performance assuming that a very large number message buffers is provided.' 'Suppose, however, that it is desired to provide only the minimum number of messages buffers required to guarantee that p k < 0.005 How many buffers should be provided?' This is a design problem. Here λ= 4/sec and μ= 4.55/sec. As required by the switching centre, we examine the minimum number of message buffers with different choices of k. Though not explicitly mentioned, it is necessary to assume reneging and balking. Balking because in telecommunication systems, it is known that an incoming message that sees a workload may be admitted in to the system with a certain probability and rejected otherwise. Rejection implies balking. We shall assume that the balking probability is 'p' (Boxma et al. 2008, has analyzed with a similar motivation).
Further we assume R_BOS because in telecommunication systems it is also known that messages usually have some real time constraints within which the message has to be processed. Messages received after the deadline is considered obsolete and discarded. This can be seen as reneging (Movaghar, 1998 andBoots andTijms, 1999). Let us assume alternative possible Markovian reneging rates of ν=0.1/sec, ν=.03333/sec. Put differently, these rates imply that a message joining the buffer will be alive if processing commences within 10sec and 30sec respectively on the average. We further assume that balking rate is independent of state and is taken as p = 0.001 (one in 1000 message).
Various performance measures of interest computed under different scenarios are given in Table 1 and 2. These measures were arrived at using a FORTRAN 77 program coded by the authors. Different choices of k were considered. Results relevant with regard to the requirement that the switching centre should provide only the minimum number of message buffers to guarantee should p k <0.005 are presented in the tables. (All rates in the following tables are per second).  In case the reneging behavior follows exp (0.1) distribution, it appears from the tables 1 that an ideal choice of k could be 14 with p k =0.00478. If the reneging distribution is exp (0.03333), then k=19 appears to be close to the switching centre with p k =.00390 (table 2).
A few interesting observations can be made from the above tables.
1) The rate of arrival into the system is 4message/sec whereas the actual load of the server (λ s ) is 3.71686 (from  table 1

Conclusion:-
The analysis of a single server finite buffer Markovian queuing system with state-independent balking and Markovian reneging has been discussed. Closed form expressions of number of performance measures have been presented. Many of these could be of direct interest to the system manger. To study the change in the system corresponding to change in system parameters, sensitivity analysis has also been presented. A numerical example with design connotations has been discussed to demonstrate results derived. An obvious extension would be the finite buffer multiserver case. Work towards that end is ongoing.