Question

explain further please

Example 6.19 Consider an M/M/1 queue in which arrivals finding N in the system do not enter. This finite capacity system can be regarded as a truncation of the M/M/1 queue to the set of states A-: {0, I, , N). Since the number in the system in the M/M/1 queue is time reversible and has limiting probabilities P(/ X/) it follows from Proposition 6.8 that the finite capacity model is also time reversible and has limiting probabilities given by 1m)

Answer 1

The service system can accommodate N customers only. The (N+1)th customer will not join the queue We know that, probability of having one customer in the queuing system is as mentioned below. -e Similarly, probability of having N customers in the service system will be Since it is finite queuing model, the sum of probabilities till Nth customer will be 1 as written below,

n=0 沁) where .-\ represents finite Geometric Progression series Hence, P, = Probability of having no customer in the finite sin gle server queue is

hence

$P_j = P_o \left ( \frac{\lambda}{\mu} \right )^j = \frac{(\frac{\lambda}{\mu})^j}{\sum_{i = 0}^{N} (\frac{\lambda}{\mu})^n}$