Consider an M/G/1/GD/∞/∞ queuing system in which interarrival times are exponentially distributed with parameter l and service times have a probability density function s(t). Let Xi be the number of customers present an instant after the ith customer completes service.

a Explain why X1, X2,…, Xk, . . . is a Markov chain.

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b Explain why Pij = P(Xk-1 = j|Xk = i) is zero for j

c Explain why for i > 0, Pi,i-1 = (probability that no arrival occurs during a service time); Pii = (probability that one arrival occurs during a service time); and for j ≥ i, Pij = (probability that j – i + 1 arrivals occur during a service time).

d Explain why, for j ≥ i – 1 and i > 0,

Hint: The probability that a service time is between x and x + x is xs(x). Given that the service time equals x, the probability that j – i + 1 arrivals will occur during the service time is


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