3 Manager's point of view

3.1 Ordinary policies

The six following policies will be tested. Policies size and load, assign the incoming customer to the queue with the smallest size or the smallest load, while rand and robn assign him either at random or cyclically (round robin). Policy rtwo, tailored for distributed environments, chooses at random two of the $n$ queues and assigns to the shorter sized one [5]. The last item, called fast, is not really an allocating policy : it describes another system where the same arrivals are treated by a single server with the same (total) capacity of service and the same coefficient of variation.

Fig. 1: Comparing six policies ($K=400$, $N=25000$)

[number of customers in the system]

[number of busy servers]

Sorting these policies by increasing efficiency, as in Fig. 1(a), one obtains : rand, robn, rtwo, size, load, while fast is far better than any other. It is clear that increasing efficiency is linked with increasing knowledge about the queuing system. The fact that size, the shortest queue policy, is not the most efficient one is well known [6,7], but the fact that efficiency is correlated with the number $n_{b}$ of busy servers is rarely emphasized.

By Little's formula, we have $\operatorname{E}_{t}\left(n_{b}\right)=n\rho$, and therefore this quantity doesn't depend on the chosen policy. Nevertheless, the policy determines heavily the probability $\rho^{*}$ of a full use of the capacity of service (Fig. 1(b)). When using fast, $\rho^{*}=\rho$ and no other policy can reach this value. Among the effective policies, load is the only one to avoid that a customer has to wait in its own line while another server is idle due to the departure of its assigned customers (exhaustivity), while size is the second better because this policy ensures at least that an incoming customer is assigned to an idle server each time that such a server exists.

3.2 Policies with jockeying

In real life, an empty line doesn't remain empty when human customers are waiting in another line : this phenomenon is called jockeying. Fig. 2 compares load with two other policies : jsiz, assignation to the shortest queue (as for size) followed by jockeying of the next to be served customer if a server becomes idle and jran, random assignation but to an idle server if any, followed by jockeying if a line becomes empty.

Fig. 2: Three policies with jockeying

[number of customers in the system]

[number of busy servers]

This graphs explains why managers aren't so much concerned with queuing organization in centralized environments : jockeying is another way to improve efficiency... in terms of clearance of the queue.

In a distributed environment, any policy based on any knowledge is difficult to enforce, due to the delays and the subsequent obsolescence of the collected information : in such a case, rtwo appears as an efficient heuristic [8].