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06q2sol

# 06q2sol - 1.203J 6.281J 13.665J 15.073J 16.76J ESD.216J...

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1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J Logistical and Transportation Planning Methods – Fall 2006 Quiz #2 Solutions Problem 1 (35 points) a). Let the states be defined by (i, j, k) where: - i is the number of Type 1 customers (i = 0, 1, 2, or 3) - j is the number of Type 2 customers (j = 0, 1, 2, or 3) - k is the type of customer being served (k = 0, 1, 2) The total number of state is 13. The state transition diagram for the system is as follows: λ λ λ 0,0,0 0,1,2 0,2,2 0,3,2 1,0,1 1,1,2 3,0,1 2,0,1 1,1,1 1,2,1 1,2,2 2,1,2 2,1,1 2 λ 2 λ 2 λ 2 µ 2 µ 2 µ 1 1 µ 1 1 1 µ 1 µ 1 µ 1 µ 2 µ 2 µ 2 µ 1 µ 1 λ 1 λ 1 λ 2 λ 2 λ 2 λ 2 λ b). In the long run, this system treats Type 1 and Type 2 customers equally. Sometimes Type 1± customers have priority and sometimes Type 2 customers do. When λ 1 = λ 2 and µ 1 = µ 2 , the± state transition diagram is perfectly symmetrical.± Therefore, L 1 = L 2 under those conditions.± c). The next customer to be served by the server is a Type 2 customers if no Type 1 customer± arrives before the completion of the current service.± The probability that a Type 1 customers arrives before the completion of the current service is± given by:±

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1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J Logistical and Transportation Planning Methods – Fall 2006 Pr( next event occurs in [ 0, ε ] and is the arrival of one Type 1 customer ) P = A Pr ( next event occurs in [ 0, ε ] ) λ P = 1 A λ 1 + λ 2 + µ 1 Therefore, the probability that the next customer to be served is a Type 2 customer is given by: λ 2 + µ 1 P = 1 P = A A λ 1 + λ 2 + µ 1 d). We start with 2 people in the system: one Type 1 customer is being served and one Type 2 customer is waiting for service. Let’s first consider that the next transition is due to an arrival of a Type 1 customer. This customer will be the next one to be served and we cannot reach the state described by the rd 3 transition. If a Type 2 customer arrives before service completion, the system is full, thus the following state transition can only be due to the end of Type 1 customer’s service. Type 2 rd customers are then given priority, and we end up in the desired state at the 3 transition if rd and only if a 3
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06q2sol - 1.203J 6.281J 13.665J 15.073J 16.76J ESD.216J...

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