For the Dupit Corp. case study introduced in Section 11.4, the management science team was able to apply a variety of queueing models by making the following simplifying approximation. Except for the approach suggested by the vice president for engineering, the team assumed that the total time required to repair a machine (including travel time to the machine site) has an exponential distribution with a mean of two hours (¼ workday).
However, the team was somewhat uncomfortable in making this assumption because the total repair times are never extremely short, as allowed by the exponential distribution.
There always is some travel time and then some setup time to start the actual repair, so the total time generally is at least 40 minutes (1/12 workday).
A key advantage of computer simulation over mathematical models is that it is not necessary to make simplifying approximations like this one. For example, one of the options available in the Queueing Simulator is to use a translated exponential distribution, which has a certain minimum time and then the additional time has an exponential distribution with some mean.
(Commercial packages for computer simulation have an even greater variety of options.)
Use computer simulation to refine the results obtained by queueing models as given by the Excel templates in the figures indicated below. Use a translated exponential distribution for the repair times where the minimum time is 1/12 workday and the additional time has an exponential distribution with a mean of 1/6 workday (80 minutes). In each case, use a run size of 25,000 arrivals and compare the point estimate obtained for Wq (the key measure of performance for this case study) with the value of Wq obtained by the queueing model.
a. Figure 11.4.
FIGURE 11.4
This spreadsheet shows the results from applying the M/M/1 model with λ = 3 and µ = 4 to the Dupit case study under the current policy. The equations for the M/M/1 model have been entered into the corresponding output cells, as shown at the bottom of the figure.

b. Figure 11.5.
FIGURE 11.5: This application of the spreadsheet in Figure 11.4 shows that, when µ= 4, the M/M/1 model gives an expected waiting time to begin service of Wq = 0.25 day (the largest value that satisfies Dupit's proposed new service standard) when l is changed from λ = 3 to λ = 2.

c. Figure 11.8.
FIGURE 11.8: This Excel template for the M/M/s model shows the results from applying this model to the approach suggested by Dupit's chief financial officer with two tech reps assigned to each territory.

d. Figure 11.9.
FIGURE 11.9: This Excel template modifies the results in Figure 11.8 by assigning three tech reps to each territory.

e. What conclusion do you draw about how sensitive the results from a computer simulation of a queueing system can be to the assumption made about the probability distribution of service times?

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M/M/1 Queueing Model for the Dupit Corp. Problem Data Results 3 4 3 (mean arrival rate) L= 4 (mean service rate) 2.25 μ - (# servers) 1 Pr(W> () = 0.368 0.75 when t = 0.75 10 P= Prob(Wt) = 11 0.276 when t= Ря 12 13 0.2500 14 0.1875 0.3000 0.1406 15 0.2500 16 3 0.1055 0.2000 17 0.0791 0.1500 18 0.0593 0.1000 19 0.0445 0.0500 6. 0.0000 20 0.0334 0 1 2 3 4 567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Number of Customers in System 21 0.0250 22 9. 0.0188 23 10 0.0141 Probability Range Name Cells L. Lambda Pr(W>t) = =EXP(-Mu*(1-Rho)*C9) 4 L= =Lambda/(Mu-Lambda) G4 C4 G5 C5 F13:F38 L= =Lambda)^2/(Mu*(Mu-Lambda) 6. Mu W = =1/(Mu-Lambda) 11 Prob(W>t) ==Rho*EXP(-Mu*(1-Rho)*C12) We= =Lambda/(Mu*(Mu-Lambda)) Po 8. G13:G38 P. Rho 10 p= =Lambda/Mu G10 C6 C9 C12 G7 G8 11 Timel Time2 12 13 =1-Rho =(1-Rho)*Rho^n =(1-Rho)*Rho'n 14 W. 15 16 3 17 4