This question is intended to build and test your intuition on the design of wait

Question

This question is intended to build and test your intuition on the design of waiting lines with variability and two identical streams of customer arrivals (e.g., men and women). The three systems being considered are shown below, where the arrow denotes flow with arrival rate rak, for stream k = 1, 2, the triangle represents the waiting line or queue and the rectangles represent processing by a server with specified average unit processing time – the two rectangles inside a larger rectangle in System B corresponds to a two-server station; System C is a single-server station with average unit process time cut by 50%; assume values for rak and te, for instance, take ra1 = ra2 = 0.8 customers per hour, te = 1 hour. Unless otherwise stated, assume all coefficients of variation are 1.1.

1.) Calculate the average time in queue in the three systems. Show your data values / approach.

Explanation / Answer

arrival rate ra1 0.8 customers/hr ra2 0.8 customers/hr avg unit processing time te 1 hour service rate 1 customer/hr average time in queue = mean arival rate / (mean service rate*(mean service rate - mean arrival rate) System A mean arrival rate = ra1=ra2= 0.8 customers/hr mean service rate = (te=1) 1 customer/hr Hence, average time in queue 4 hours System B mean arrival rate = ra1+ra2= 1.6 customers/hr mean service rate = (te =1+1)= 2 customer/hr (both serving 1 customer/hr hence collectively 2 customers/hr) Hence, average time in queue 2 hours System C mean arrival rate = ra1+ra2= 1.6 customers/hr mean service rate (te= 1/2) 2 customer/hr (1 ustomer in half hour, hence 2 customers/hour) Hence, average time in queue 2 hours

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