Single server queuing model

Single server queuing model

Single server queuing Model 1: (MM1) : (   / FIFO)Image result for Single server queuing model in Operation Research

This model is based on the following assumptions:
(i) The arrivals follow Poisson distribution, with a mean arrival rate .
(ii) The service time has exponential distribution, average service rate .
(iii) Arrivals are infinite population a.
(iv) Customers are served on a First-in, First-out basis (FIFO).
(v) There is only a single server.

System of Steady-state Equations

In this method, the question arises whether the service can meet the customer demand. This depends on the values of and.
If , i.e., if arrival rate is greater than or equal to the service rate, the waiting line would increase without limit. Therefore for a system to work, it is necessary that <.
As indicated earlier, traffic intensity = / . This refers to the probability of time. The service station is busy. We can say that the probability that the system is idle or there are no customers in the system, P0 = 1 – .
From this, the probability of having exactly one customer in the system is P1 = P0.
Likewise, the probability of having exactly 2 customers in the system would be P3 = P1 = 2 P0
The probability of having exactly n customers in the system is
Pn = nP0 = n(1-r) = ( / )n P0