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A. A computer rejuvenation facility receives systems from the fleet at 24 systems/day (assume this is a Poisson process).  It takes the triage function 30 minutes (assumed exponentially distributed) to assess the refurbishment requirements of a system.  40% of the systems are sent to keyboard replacement, where repairs are done by a lone service agent who takes 60 minutes (exponentially distributed)  to replace the keyboard.  The other 60% are put in the dumpster.

1. Draw the Jackson network.

2. Give R, the routing matrix (you can include the “0” station if you like, your choice)

3. Determine lambda, L, Lq, W, Wq for the keyboard replacement station.

4. What is the distribution of the time between arrivals at the keyboard replacement station?

5. The facility experiences a budget cut, causing the triage processor to slow down because his teeth hurt (his dental insurance got cut!)  How slow can the triage agent become before his speed becomes a problem (slows the system down in some way)?

6. What is the total time it takes a typical computer to process through the entire system, assuming it is not junked? The dumpster holds 10 computers, and is emptied once per day (deterministic).  What is the probability it overflows…

7. On a random day?

8. Sometime during a random week?

B. Northbound trucks leaving Helmand Province with USMC communications equipment must go through a washing process before getting underway, so as to prevent Tajikistan from getting “too sandy.”  Trucks are ready to wash at a sustained rate of one every 2 hours (exponentially distributed), and a washdown takes 4 hours (exponentially distributed).  There are 10 wash stations that operate in parallel, with 10 wash technicians (hosers) manning the wash stations.  Whenever all 10 wash stations are simultaneously idle, the hosers smoke a communal hookah together until the next truck arrives for a wash.

1. This is a X/Y/Z/W kind of queue (give X, Y, Z, and W).

2. Give rho for this system.

3. Is this queue stable?

4. On average, how long do the hosers smoke the hookah?

5. On average, how many hookah smoking breaks to the hosers enjoy per day?

6. On average, how long is the work period between smokes?

7. On average, how many trucks get washed between smokes?

C. Summon up the model SENSITIVITY OF M/G/1 TO THE SERVICE-TIME COEFFICIENT OF VARIATION in QTS.  Recall that CoV = stddev(service time)/mean(service time), the ratio of the service time’s standard deviation to its mean.  Make rho = .9, and run the model.

1. For CoV = 1.0, L should be 9.0.  Why does this make sense?

2. For CoV = 0, how does this queue operate?

3. For CoV’s other than 1.0, does Little’s Law apply?  If so, why isn’t the L = 9.0 for all values of CoV?

4. find out L-Lq for several values of CoV.  Comment.

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