Page 185 - Contributed Paper Session (CPS) - Volume 3

CPS1988 Saggou Hafida H. et al.
The paper is structured as follows. In the next section, we give the
mathematical description of our queueing model. In section 3, we analyze the
steady-state distribution of the queueing system under consideration. The
different probabilities and important performance measures and reliability
indices this model are given in section 4. In section 5, we present the stochastic
decomposition property.

2. The mathematical model
We consider an [] //1 retrial queueing model with two types of
customers: transit (also called ordinary) customers and a fixed number K (K ≥
1) of recurrent (also called permanent) customers. Our model is based on the
following hypotheses.
1. Transit Customers arrives at the system according to a compound Poisson
process with rate λ. Let X be the Upon arrival, if a batch of transit customers
finds the server busy, it can joins the retrial group (orbit) in accordance
with an F.C.F.S discipline with probability p or leaves the system with
probability 1 − . We assume that only the transit customer at the head of
the orbit is allowed to access the server. Successive inter-retrial times of
any transit customer follow an arbitrary probability distribution function
(), with a corresponding density function () and a Laplace-Stieldjes
transform LA().

2. A single server can serve only one-by-one transit customer at a time based
on the F.C.F.S discipline of service concerning the batch transit arrival.
Successive service times are independent with a common probability
distribution function (), a density function (), a Laplace-Stieltjes
1
1
th
Transform (LST) (), n moments .
1
1

3. There is a fixed number of recurrent customers in the system. Once
served, recurrent customers immediately return to the retrial group in
accordance with an F.C.F.S discipline. We assume that only the recurrent
customer at the head of the orbit is allowed to access the server. The
recurrent customer at the head of the group repeats his call after an
amount of time following an exponential distribution with the parameter
.

4. The service time of recurrent customers are i.i.d with a probability
distribution function (),, a density function (),), a Laplace-Stieldjes
2
2
th
transform () and n moments
2
2

5. Once the service time of the transit customer is completed, the server can
go on vacation with probability 1 − or stays in the system for serving
174 | I S I W S C 2 0 1 9