The engines on jet aircraft must be periodically inspected and, if necessary, repaired. An...

The engines on jet aircraft must be periodically inspected and, if necessary, repaired. An inspection/repair facility at a large airport handles seven different types of jets, as described in the table below. The times between successive arrivals of planes of type i (where i = 1, 2, …, 7) are exponentially distributed with mean a(i), as given in the table; all times are in days. There are n parallel service stations, each of which sequentially handles the inspection and repair of all the engines on a plane, but can deal with only one engine at a time. For example, a type 2 plane has three engines, so when it enters service, each engine must undergo a complete inspection and repair process (as described below) before the next engine on this plane can begin service, and all three engines must be inspected and (if necessary) repaired before the plane leaves the service station. Each service station is capable of dealing with any type of plane. As usual, a plane arriving to find an idle service station goes directly into service, while an arriving plane finding all service stations occupied must join a single queue.

Two of the seven types of planes are classified as widebody (denoted by an asterisk * in the above table), while the other five are classified as regular. Two disciplines for the queue are of interest:

(i) Simple FIFO with all plane types mixed together in the same queue

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(ii) Nonpreemptive priority given to widebody jets, with the rule being FIFO within the widebody and regular classifications

For each engine on a plane (independently), the following process takes place (i denotes the plane type):

• The engine is initially inspected, taking an amount of time distributed uniformly between A(i) and B(i).

• A decision is made as to whether repair is needed; the probability that repair is needed is p(i). If no repair is needed, inspection of the jet’s next engine begins; or if this was the last engine, the jet leaves the facility.

• If repair is needed, it is carried out, taking an amount of time distributed as a 2-Erlang random variable with mean r(i).

• After repair, another inspection is done, taking an amount of time distributed uniformly between A(i)/2 and B(i)/2 (i.e., half as long as the initial inspection, since tear-down is already done). The probability that the engine needs further repair is p(i)/2.

• If the initial repair was successful, the engine is done. If the engine still fails inspection, it requires further repair, taking an amount of time distributed as 2-Erlang with mean r(i)/2, after which it is inspected again, taking an amount of time distributed uniformly between A(i)/2 and B(i)/2; it fails this inspection with probability p(i)/2, and would need yet more repair, which would take a 2-Erlang amount of time with mean r(i)/2. This procedure continues until the engine finally passes inspection. The mean repair time stays at r(i)/2, the probability of failure stays at p(i)/2, and the inspection times stay between A(i)/2 and B(i)/2.

A cost of c(i) (measured in tens of thousands of dollars) is incurred for every (full) day a type i plane is down, i.e., is in queue or in service. The general idea is to study how the total (summed across all plane types) average daily downtime cost depends on the number of service stations, n. Initially the system is empty and idle, and the simulation is to run for 365 round-the-clock days. Observe the average delay in queue for each plane type and the overall average delay in queue for all plane types, the time-average number of planes in queue, the time-average number of planes down for each plane type separately, and the total average daily downtime cost for all planes added together. Try various values of n to get a feel for the system’s behavior. Recommend a choice for n, as well as which of the queue disciplines (i) or (ii) above appears to lead to the most cost-effective operation. Use streams 1 through 7 for the interarrival times of plane types i = 1 through i = 7, respectively, streams 8 through 14 for their respective inspection times (first or subsequent), streams 15 through 21 to determine whether they need (additional) repair, and streams 22 through 28 for their repair times (first or subsequent).

FIGURE 2.71 Alternative layout for the aircraft-repair facility.

As an alternative to the above layout, consider separating entirely the service of the widebody and regular jets. That is, take n₂ of the n stations and send all the widebody jets there (with a single queue of widebodies feeding all n₂ stations), and the remaining n₁ = n – n₂ stations are for regular jets only; see Fig. 2.71. Do you think that this alternative layout would be better? Why? Use the same parameters and stream assignments as above.