Exercise 7 M/M/ · · · Queueing- and Loss-Systems 1. A pure loss system with negativ exponentially distributed inter-arrival times and service times has exactly n service units. Determine the probability of exactly k = 0 , . . . ,n service units being occupied. What will be the traﬃc level and the loss probability for this station? 2. A processor is connected to the receiving end of a data transmission line. The processor is able to buﬀer K symbols from this line. The arrival process on this line is a Poisson process with the rate λ , and the service time needed for each symbol is negative exponetially distributed with the completion rate μ . Symbols are lost if they meet a full buﬀer at arrival. Calculate the state probabilities p k , the waiting probability p w , the loss probability p b and the traﬃc level Y with K = 2 buﬀer places and a traﬃc load of ρ = 1. 3. Consider a queueing loss system M/M/n–s/FCFS with n service units and s buﬀer places. (a) Show the state space with all state transitions and transition probability densi-
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Probability theory, communication networks, service units, state probabilities, state probabilities pk