dolution

Consider the finite queue variation of the M/G/1 model, where K is the maximum number of customers allowed in the system. For n = 1, 2, . . . , let the random variable X_n be the number of customers in the system at the moment t_n when the nth customer has just finished being served. (Do not count the departing customer.) The times {t₁, t₂, . . .} are called regeneration points. Furthermore, {X_n} (n = 1, 2, . . .) is a discrete time Markov chain and is known as an embedded Markov chain. Embedded Markov chains are useful for studying the properties of continuous time stochastic processes such as for an M/G/1 model.

Now consider the particular special case where K = 4, the service time of successive customers is a fixed constant, say, 10 minutes, and the mean arrival rate is 1 every 50 minutes. Therefore, {X_n} is an embedded Markov chain with states 0, 1, 2, 3. (Because there are never more than 4 customers in the system, there can never be more than 3 in the system at a regeneration point.) Because the system is observed at successive departures, X_n can never decrease by more than 1. Furthermore, the probabilities of transitions that result in increases in X_n are obtained directly from the Poisson distribution.

(a) Find the one-step transition matrix for the embedded Markov chain. (Hint: In obtaining the transition probability from state 3 to state 3, use the probability of 1 or more arrivals rather than just 1 arrival, and similarly for other transitions to state 3.)

(b) Use the corresponding routine in the Markov chains area of your IOR Tutorial to find the steady-state probabilities for the number of customers in the system at regeneration points.

(c) Compute the expected number of customers in the system at regeneration points, and compare it to the value of L for the M/D/1 model (with K = ∞) in Sec. 17.7.