Queuing Analysis

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 Process Flow Analysis - Queuing Analysis

The Process Economics worksheet includes a queuing analysis for the resources defined for system. The analysis is shown below for the example problem.

This display is to the right of the resource analysis. Since the formulas used in the queuing analysis refer to columns from the resource analysis, we repeat it below for the example.

 Range Title Explanation Q9:Q12 Queuing Analysis Column Q lists the resources of the system. R9:R12 Arrival Rate per Week Column R holds the arrival rate into each resource. This is the cumulative number of arrivals for all products and all operations that use the resource. It is expressed in terms of the system time interval (Week in this case) S9:S12 Service Rate per Week Column S holds the service rate of each resource. The Excel formula for the service rate for the Pack resource is: service rate for Pack =R9*I9*H9/N9 Service Rate = [arrival rate][hours per week]*[availability of the resource]/[total time used on this resource] For the Pack operation this yields (rounded to integers): Service rate for Pack =(1734/week)*(40hrs./week)*(1)/(104hrs./week) = 667/week T9:T12 Number of Servers The number of servers in column T is transferred from column F of the resource analysis. U9:U12 Average Time in Queue The average queue time in column U is computed with the function pf_Wq provided by the add-in. This function implements the queuing formulas for a station with Poisson arrival and service processes. Since the arrival and service rates are per week, the result is multiplied by the content of H9 which translates the results to the operating interval (hours in this case). Average queue time for Pack = pf_Wq(R9,S9,T9)*H9 = (0.0029 weeks)*(40 hr./week) = 0.1146 hours V9:V12 Average Number in Queue The average number in queue is the average time in queue multiplied by the arrival rate. For Pack we compute Average number in queue for Pack = R9*U9/H9 = (1734/week)*(0.1146 hours)/(40 hr./week)= 4.97

The queuing analysis requires a number of assumptions. First we assume that items using a resource visit individually and in random order. Further we assume that the interarrival times come from an exponential distribution and that the service times come from an exponential distribution. The means of the distributions are computed from the process data. The operation times for the items include an allowance for setup times, if that parameter is used, but the items are not processed in a lots. Lot processing would invalidate the assumptions of exponential service time. If these assumptions make the analysis invalid, the queuing results can simply be neglected.

The results do provide estimates of delays due to queues waiting for resources, and queues often occur in a manufacturing situation. The queue times could be used as delay times in the process definitions. Delays do not affect resource utilization, but they do affect throughput time. An analysis using the queuing results could be more accurate than an analysis that does not.

All the cells of the queuing analysis are implemented with formulas, so no interaction is necessary from the user. In cases when the capacity of the resource is equal or less than the use of the resource, the queuing analysis will indicate an error. The queuing formulas are not valid in these cases, and in fact, queues are infinitely long. Of course an optimum solution obtained with a linear mathematical programming model will use all bottleneck resources to their fullest. We use the Max. % utilization in column J of the data to limit the use to the percentage shown here. If one sets this number less than 100%, the resource utilization will always be less than 1 for an optimum solution and queues will be bounded.

Note that the analysis here is different than the analysis provided by queues within a process. For the latter case, the flow through the queue is entirely the flow for the process, and not the combination of flows from several processes. The results for the queue within a process do depend on the lot size.

Operations Management / Industrial Engineering
Internet
by Paul A. Jensen