S-38.149

Extensions to a queue with a single server

Excercises to be returned on Mon 9 Nov

Markus Isomäki

1.Consider a modem pool with total c customers, m available modems and w waiting places. The calls of each customer are on the average separated by 1/l seconds, and this time is assumed to be exponentially distributed. The calls last on the average 1/m seconds, and this time is also assumed to be exponentially distributed. A customer who would be placed at the waiting place will not enter the system at the probability r_n, where n is the number of waiting (not in service) customers in front of him, e.g. if the customer is first to wait, he will depart at probability r₀. After a departure or a blocked attempt, the customer calls in again after 1/l seconds on the average (exponential distribution), so departure and blocking do not affect his behaviour.

• (a) Draw the Markov chain of the system and determine the equilibrium equations.
• (b) Let c=10, m=4, w=2, 1/l =500, 1/m =100, r₀=0.2, r₁=0.6. Calculate the probability that an arriving customer does not get the service, e.g. he is blocked or departs.

2. Consider a M^x/M/1 queue with batch arrivals.Customers arrive to the system in groups of 1, 2 and 3. The total arrival rate l =10 groups/second, and the probabilities of the group sizes 1, 2, and 3 are C₁=0.5, C₂=0.3 and C₃=0.2, respectively. Draw the Markov chain of the system, determine the equilibrium equations, use the generating polynomials to determine the state probabilities and calculate numerical results for p(0), p(1), p(2) and p(3).