A single server compulsory vacation queue with three type of services and with restricted admissibility

queue when the server is on vacation, the probability generating function of the number of customers in the queue when the server provides th i type of service ) 3 , 2 , 1 = (i and the probability generating function of the number of customers in the queue irrespective of the server states are derived in section 4. Also, some performance measures related to this queueing model are obtained from these probability generating functions and are given in section 5. In sections 6 and 7, particular model is derived by assigning particular values to the parameters and a numerical study is carried out respectively.

Model Description

Consider a single server queue with the customers arrival follows, according to a Poisson process of intensity λ and the server provides two types of services, respectively called type 1 service and type 2 service. At the beginning of a service, it is assumed that the customer has the choice of selecting type 1 service with probability 1 p or type 2 service with probability

The type 1 service is a phase type service (two phases). After completion of type 1 service, the customer leaves the system, whereas after completion of type 2 service, the customer leaves the system with probability 3 -1 p or choose an optional service with probability .

3 p After completion of optional service, the customer leaves the system. Here the optional service is called type 3 service. The service discipline is assumed to be first come, first served (FCFS). The service time distributions are general, the distribution functions are ) (

After completion of each service the server takes vacation for a random period of time, called vacation period. This vacation period V is independently and identically distributed with distribution function ), (x V Laplace-Stieltjes transform (LST)

Further, it is assumed that not all the arriving customers are allowed to join the system at all time. Let r

r be the probability that an arriving customer will be allowed to join the system while the server is busy or idle and let p ) 1 < < 0 ( p be the probability that an arriving customer will be allowed to join the system while the server is on vacation. It may be noted that

and are the first order differential functions (hazard rates) of

and Q denote the corresponding steady state probabilities.

The probability generating functions for the probabilities     



be the conditional probability of completion of the given that the elapsed vacation time is .

The Governing Equations

We utilize the argument of Cox [1] and obtain the following Kolmogorov forward equations related to the above defined model under the steady state conditions:

The above set of equations is to be solved under the following boundary conditions at

and the normalization condition is  1

The Analysis

Multiplying equation (2) by , n z summing from 1 = n to  and then adding (1), we get

Integration of the equation (20) leads to

Taking x = 0 in equation ( 21), the constant C is obtained as

Using equation ( 33) in ( 28) and (30), we get

Using equations (34), ( 35) in ( 29) and (31), we get

Integration of equations ( 23)-( 27) with respect to x and using (33)-(37), we get

in equations ( 38)-(42) and using equation ( 19) leads to

Equation ( 43) is called the probability that the server is idle and no one in the system. Equations ( 38)-( 42) together with equation ( 43) are respectively, the probability generating functions of the number of customers in the queue when the server is serving type 1 service and is in the th j ) 2 , 1 = ( j phase of service, serving type 2 service and type 3 service respectively, the server is on vacation. Here 0  Q guarantees the existence of the probability generating functions in equations ( 38)-( 42) and therefore the stability condition for the system is

The probability generating function that the number of customers in the queue irrespective of the server state is

The Performance Measures

Using straightforward calculations the following performance measures have been obtained: (i) The mean number of customers in the queue is

is given in equation ( 44). (ii) The mean waiting time in the queue is

The results coincides with the result of Madan [14] .

Particular Model

In this section, a particular model has been calculated by taking known distribution to service time and vacation time. The service times distribution are negative exponential with parameters ). ( 21 q q  The following performance measures have been calculated.

The Numerical Study

In this section, numerical results are pressented related to the model discussed in section 6. We fix some parameter values , 0 .  has been varied from 1.1, 1.5, 1.9, ( 1 has been varied from 1.1, 1.5, 1.9)). Figures 1,2 represents the mean number of customers in the queue whereas 3, 4 indicates that the mean waiting time in the queue with respect to arrival rate. All the functions are increasing function. Table 1 represents the probability that the server is idle. From the table it is clear that the arrival rate increases, the corresponding probability is also increases.

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