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12.5 Constant Service Time Model (M /D / 1)

Some service systems have constant service times instead of exponentially distributed times. When customers or equipment are processed according to a fixed cycle, as in the case of an automatic car wash or an amusement park ride, constant service rates are appropriate. Because constant rates are certain, the values for $L_{q}, W_{q}, L,$ $L_{q}, W_{q}, L,$ and W are always less than they would be in the models we have just discussed, which have variable service times. As a matter of fact, both the average queue length and the average waiting time in the queue are halved with the constant service rate model.

Equations for the Constant Service Time Model

Constant service model formulas follow:

Average length of the queue:

$L_{q} = \frac{λ^{2}}{2 μ (μ - λ)}$ $L_{q} = \frac{λ^{2}}{2 μ (μ - λ)}$ (12-19)
Average waiting time in the queue:

$W_{q} = \frac{λ}{2 μ (μ - λ)}$ $W_{q} = \frac{λ}{2 μ (μ - λ)}$ (12-20)
Average number of customers in the system:

$L = L_{q} + \frac{λ}{μ}$ $L = L_{q} + \frac{λ}{μ}$ (12-21)
Average time in the system:

$W = W_{q} + \frac{1}{μ}$ $W = W_{q} + \frac{1}{μ}$ (12-22)

In Action Queuing at the Polls

The long lines at the polls in recent presidential elections have caused some concern. In 2000, some voters in Florida waited in line over 2 hours to cast their ballots. The final tally favored the winner by 527 votes of the almost 6 million votes cast in that state. Some voters growing tired of waiting and simply leaving the line without voting may have affected the outcome. A change of 0.01% could have caused different results. In 2004, there were reports of voters in Ohio waiting in line 10 hours to vote. If as few as 19 potential voters at each precinct in the 12 largest counties in Ohio got tired of waiting and left without voting, the outcome of the election may have been different. There were obvious problems with the number of machines available in some of the precincts. One precinct needed 13 machines but had only 2. Inexplicably, there were 68 voting machines in warehouses that were not even used. Other states had some long lines as well.

The basic problem stems from not having enough voting machines in many of the precincts, although some precincts had sufficient machines and no problems. Why was there such a problem in some of the voting precincts? Part of the reason is related to poor forecasting of the voter turnout in various precincts. Whatever the cause, the misallocation of voting machines among the precincts seems to be a major cause of the long lines in the national elections. Queuing models can help provide a scientific way to analyze the needs and anticipate the voting lines based on the number of machines provided.

The basic voting precinct can be modeled as a multichannel queuing system. By evaluating a range of values for the forecast number of voters at different times of the day, a determination can be made on how long the lines will be, based on the number of voting machines available. While there still could be some lines if the state does not have enough voting machines to meet the anticipated need, the proper distribution of these machines will help to keep the waiting times reasonable.

Source: Based on Alexander S. Belenky and Richard C. Larson, “To Queue or Not to Queue,” OR/MS Today 33, 3 (June 2006): 30–34, © Trevor S. Hale.

Garcia-Golding Recycling, Inc.

Garcia-Golding Recycling, Inc., collects and compacts aluminum cans and glass bottles in New York City. Its truck drivers, who arrive to unload these materials for recycling, currently wait an average of 15 minutes before emptying their loads. The cost of the driver and truck time wasted while in queue is valued at $60 per hour. A new automated compactor can be purchased that will process truck loads at a constant rate of 12 trucks per hour (i.e., 5 minutes per truck). Trucks arrive according to a Poisson distribution at an average rate of 8 per hour. If the new compactor is put in use, its cost will be amortized at a rate of $3 per truck unloaded. A summer intern from a local college did the following analysis to evaluate the costs versus benefits of the purchase:

\begin{array}{l} C u r r e n t waiting cost/trip & = & (\frac{1}{4} hour waiting now) ($60/hour cost) \\ = & $15/trip \\ N e w system: λ & = & 8 trucks/hour arriving \\ μ & = & 12 trucks/hour served \\ Average waiting time in queue & = & W_{q} = \frac{λ}{2 μ (μ - λ)} = \frac{8}{2 (12) (12 - 8)} \\ = & \frac{1}{12} hour \\ Waiting cost/trip with new compactor & = & (\frac{1}{12} hour wait) ($ 60 / hour cost) = $ 5 / trip \\ Savings with new equipment & = & $15 (current system) - $5 (new system) \\ = & $10/trip \\ Cost of new equipment amortized & = & $3/trip \\ Net savings & = & $7/trip \end{array}

$\begin{array}{l} C u r r e n t waiting cost/trip & = & (\frac{1}{4} hour waiting now) ($60/hour cost) \\ = & $15/trip \\ N e w system: λ & = & 8 trucks/hour arriving \\ μ & = & 12 trucks/hour served \\ Average waiting time in queue & = & W_{q} = \frac{λ}{2 μ (μ - λ)} = \frac{8}{2 (12) (12 - 8)} \\ = & \frac{1}{12} hour \\ Waiting cost/trip with new compactor & = & (\frac{1}{12} hour wait) ($ 60 / hour cost) = $ 5 / trip \\ Savings with new equipment & = & $15 (current system) - $5 (new system) \\ = & $10/trip \\ Cost of new equipment amortized & = & $3/trip \\ Net savings & = & $7/trip \end{array}$

Using Excel QM For Garcia-Golding’s Constant Service Time Model

To use Excel QM for this problem, from the Excel QM menu, select Waiting Lines - Constant Service Time Model (M/D/1). When the spreadsheet appears, enter the arrival rate (8) and the service rate (12). Once these are entered, the solution shown in Program 12.3 will be displayed.

A screenshot of a spreadsheet shows the solution for Constant Service Time Model for Garcia-Golding Recycling Example. — Program 12.3 Excel QM Solution for Constant Service Time Model for Garcia-Golding Recycling Example

Figure 12.3 Full Alternative Text

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Table of Contents for 12.5 Constant Service Time Model (M /D / 1)

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Table of Contents for
12.5 Constant Service Time Model (M /D / 1)