14.4 Designing a service system

A typical decision is the number of servers. If a server costs \(C_s\) an hour and a customer’s time in the system costs \(C_w\) an hour, the hourly cost with \(c\) servers is \(cC_s + C_wL(c)\), where \(L(c) = L_q(c) + a\); the best \(c\) is found by computing \(L(c)\) from the Erlang C formula for a few values of \(c\) (Problem 14.6.5). Service-level targets, such as answering \(80\%\) of calls within \(20\) seconds, are handled the same way, using the waiting-time distribution \(P(W_q > t) = C(c, a)e^{-(c\mu - \lambda )t}\). For large systems the square-root staffing rule \(c \approx a + \beta \sqrt a\) achieves a fixed service level, so that the spare capacity needed grows only like the square root of the load: large systems are more efficient, the economies of scale of pooling. Networks of queues, in which customers move between service stations, have product-form solutions when arrivals are Poisson and services exponential (Jackson, 1957); otherwise they are studied by simulation, the subject of the last section.

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