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n s 1 p sp n E 0 1 p s p 1 Sp N...

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Algebra

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n s 1 p sp n E 0 1 p s p 1 Sp N s 1 p 1 This queueing modrl with limited waiting room is valuable because of its relevance to many real situations and the fact that changes may be made to its properties by adjusting the number of servers or the capacity of the waiting room However while poission arrivals are common in practice negative exponential service times are less so and it is the second assumption in the system MIMs that limits its usefulness Example A supermarket has two girls ringing up stales at the counters If the service time for each customer is exponential with mean 4 minutes and if people arrive in a poission fashion at the counter at the rate of 10 per hour then calculate Solution a The probability of having to wait for service b The expected percentage of idle time for each girl c If a customer has to wait find the expected length of his waiting time a Probability of having to wait for service is P W 0 Po s 1 p 1 6 s s 2 Now compute Po 1 n 0 52 16 0 5 p s sp n sp s n n s 1 P n 0 2 1 3 2 1 3 1 2 3 4 9 2x2 31 1 2 Thusprob W 0 21 1 3 1 6 b The fraction of the time the service remains busy is given by p us 1 3 2 1 3 1 3 Theefore the fraction of the time thee service remains idle is 1 1 3 2 3 67 nearly c W W 0 3 minutes

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