5 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.

b Explain why Pij P(Xk1 j|Xk i) is zero for j i 1.

c Explain why for i 0, Pi,i1 (probability that no arrival occurs during a service time); Pii (probability that one arrival occurs during a service time); and for j

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i, Pi (probability that ji+1 arrivals occur dur- ing a service time). d Explain why, for ji - 1 and i > 0, Pij [j-i s(x)e^x(x)+1 dx Hint: The probability that a service time is between x and x + Ax is Axs(x). Given that the service time equals x, the probability that ji+1 arrivals will occur during the service time is e-Ax (x)+1 (j-i + 1)!