As a second example of a queueing network, consider the job shop manufacturing system shown in the figure below. The rectangles identify machine types. There are three classes of products which take different routes through the shop. The order rate for the classes along with the routing information is shown in the table. We assume that the orders for the product arrivals are Poisson processes with the rates given and that processing times at the machines have exponential distributions. |

The queueing network model constructed by the add-in is shown below. Since all products first pass through station A, the independent arrival rate at that station is the sum of the order rates for the three products. The proportions transferred from one station to the next for the example are shown in the Transfer matrix. The resultant arrival rates at the machines are computed and shown in row 4. Service rates are given for each machine and entered into row 5. We see in row 6 the minimum number of machines necessary to carry the load. The results for the each machine type are computed in rows 7 through 14, with the totals for the system shown in column J. |

Analysis of the results show that machines B and F are the most highly utilized. The mean number in the system shown in cell J8 is the number of all four products. To compute the number of an individual product, we must first find the throughput time for the product by adding the station times for the machines through which the products pass. For example product 1 passes through machines A, B, E and F. Adding the mean times for each of these stations we find the throughput time for product 1, that is 0.789 months. Multiplying this by the flow rate of the product, which is 30 per month, we find the mean number of product 1 in the system, 23.67. This is the Work-in-Process, or WIP, for the product. Many kinds of analysis of this system are possible using the queueing network model. |

Operations
Research Models and Methods

by Paul A. Jensen and Jon Bard, University of Texas, Copyright
by the Authors