State of the art in the optimal number of servers

As early as 1958, Morse observed that for an M/M//FCFS system, the optimal number of servers is one () [131]. In 1970, Stidham formalized and generalized the observation by Morse[188]; Stidham showed that a single server is optimal in a G/Er//FCFS system, namely when the service demand has an Erlang distribution, including an exponential distribution, or is deterministic, for a general arrival process. Stidham's result is extended by Brumelle to the system where the service demand is at most as variable as the exponential random variable; i.e., a single server is optimal in a G/G//FCFS system where the service demand distribution has the squared coefficient variation at most one () [31]. For general service demand distributions, only results for limiting cases are known. Specifically, the work by Reiman and Simon [161] can be used to show that a single server () is optimal in the light traffic limit (when servers are almost always idle) [120], and the work by Iglehart and Whitt [79] implies that a multiserver system (GI/GI//FCFS) and its single server counterpart (GI/GI/1/FCFS) coincides in the heavy traffic limit (when servers are almost always busy).

All the above work proves that a single server () is the best for
special cases. However, Stidham
also discusses that the optimality of a single server
does not appear to hold for highly variable job size distributions
[188].
Very recently, Molinero-Fernandez et. al. [130]
consider the question of how many servers are best in an M/HT/
single priority system, where HT denotes a *heavy-tailed* service
distribution. To answer this question, they approximate a
heavy-tailed distribution with a bimodal distribution, and
observe that in general multiple servers are better than a single
server.

In summary, although the question of ``how many servers are best?'' has been addressed in a number of papers in the context of FCFS, results are limited to special cases such as light and heavy traffic limits and low variable and heavy tailed distributions. This motivates us to further study this question under a wide range of loads and job size variabilities in the context of FCFS before studying priority systems.