A large manufacturing concern with a 100 acre plant had a well established medical facility, which was located at the plant offices at the eastern edge of the property. Over the years, the plant had grown from east to west and travel time to the medical facility was so great that management was considering dividing the facility. The second unit was to be established near the center of the west end of the plant. A study had been made of weighted travel times for the existing single facility and for the proposed two facility system. The result indicated that average travel time for the existing large medical facility was 15 minutes. The volume averaged 1000 visits per week or 250 worker hours for travel time. The two facility plan would reduce the average travel time to 8 minutes, or 133 worker hours per week.
The question was what would happen to waiting time in the waiting rooms? For the one large facility, ג = 25 per hour, and average service time was 20 minutes, or a service rate of µ = 3 per hour, and r= 25 / 3 = 8.33. There were 10 physicians who handled the load. Interpolating Lq = 2.45, and Wq= 2.45 x 60 / 25 = 5.88 minutes per person or 98 worker hours per week. Therefore, the travel time plus the waiting time was 250 + 98 = 348 worker hours per week.
The plan was to divide the medical staff for the two facilities, and it was assumed that the load would divide equally, so comparable data for the divided facilities were ג = 12.5 per hour per facility, µ =3 per hour, M = 5, and r = 4.2. From the table of the Appendix, Lq = 3.3269 and Wq = 16 minutes per person or 267 worker hours per week. The travel plus waiting time for the dual facility plan was therefore 133 + 267 = 400 worker hours per week, compared with 348 for the single large facility. Other alternatives could be compared, probably involving an increased medical staff.
The waiting time for the single large facility was 5.88 minutes per person compared with 16.0 minutes per person for the two facility plan. The large facility gives better service than the two smaller facilities. If we visualize the two decentralized facilities functioning side by side, we can see intuitively why waiting time increases. If facility, 1 were busy and had patients waiting while at the same time facility 2 happened to be idle, someone from the facility 1 waiting room could be serviced immediately by facility 2, there by reducing the average waiting time. In this situation the two facilities are drawing from one waiting line. When they are physically decentralized, the facilities must draw on two independent waiting lines and the idle capacity of one cannot be used by the waiting patients of the other.
Cost and capacity in waiting line models:
Although many decisions concerning service systems may turn on the physical factors of line length, waiting time, and service facility utilization, very often system designs will depend on comparative costs for alternatives. The costs involved are commonly the costs of providing the services versus the waiting time costs. In some instances the waiting time costs are objective, as when the enterprise employs both the servers and those waiting. The company medical facility just discussed is such a case. The company absorbed all the travel time and waiting time costs as well as the cost of providing the service. In such an instance, a direct cost minimizing approach can be taken by balancing the waiting costs, or the time in system costs, against the cost of providing the service.
When the arriving units are customers, clients, or patients, the cost of making them wait is less obvious. If they are customers, excessive waiting may cause irritation and the loss of goodwill and eventually sales. Placing a value on goodwill, however, is not straightforward exercise. In public service operations and other monopoly situations, the valuation of waiting cost may be even more tenuous because the individual cannot make alternative choices. In these situations where objective costs cannot be balanced, it may be necessary to set a standard for waiting time, for example, to adjust capacity to keep average waiting time at or below a given number of minutes at supermarket checkout counters.