Queuing Add-in - Closed Queuing Networks
 Figure 1 A closed queuing network consists of several stations as illustrated in Fig. 1. For this structure the flow through each station is passed to other stations. It is a closed network because no flow enters from outside the network. The number of items present in the system is N. Each component of the network is a single server station with the service activity represented by a circle and the queue represented by the delay symbol (rectangular on one end and curved on the other). Each station has a specified processing rate with a mean processing time equal to the inverse of the rate. Processing times are assumed to have exponential distributions. Items are served with a first-come-first-served discipline. When an item is completed at one station it passes to other stations according to transition probabilities given by the numbers on the arcs connecting the stations. The analysis of the closed system is based on mean value analysis. For a description of the analysis, see the pdf supplement, Closed Queuing Systems. To add a model for a closed network of queues, place the cursor at the worksheet cell where the model is to be located and select Closed Network from the menu. The dialog box below allows entry of the location, name and the number of stations in the system. The worksheet for the example is shown below. We will not try to explain on these pages how the analysis is performed (see the supplement). Here we describe the contents of the cells for the closed system. The analysis assumes each station has a single server. The only data that the user must enter are the station names in row 1, station service rates in row 4, and the transition probabilities in rows 16 to 18. The other cells on the display are either yellow, indicating a formula or fixed value, or green, indicating the results of an algorithmic computation. The rows of the display are described by the titles in column A. The initial display shows the results when N is equal to 1. Columns A through D show station results and columns E and F show system results. TTo obtain a solution for N > 1, select Solve Closed from the menu. The dialog below appears. Enter the name of the network to be analyzed. The Max. Population entry is the maximum value of N. Mean value analysis is an iterative procedure that computes results for N = 1, 2, etc. up to this maximum value. The entry for Max. Throughput is provided so that the analysis will stop when the specified throughput is reached. The analysis stops when either the Max. Population or the Max. Throughput is reached. When the Show Iterations button is checked, the results for each value of N are shown below the network display. The results for N equal to 10 is below. The table below shows the iteration results for the example with the value of N ranging from 1 to 10. As the number in the system increases, the throughput also increases, but at a slower and slower rate. For the higher values, the congestion in the system causes the system time to increase as more and more items spend time in the queues. The analysis is valid only for the data present when the Solve Closed procedure is called. If the data is changed, the results are no longer valid and the procedure must be called again.

Operations Research Models and Methods
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by Paul A. Jensen
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