5 Dependence on global variances

We will now examine what happens when the customer flow increases and the number of servers increases accordingly. When the flow is not Poisson, the way that this increase is modeled has to be specified. In any case, the very fact of pooling has a strong influence over the sojourn time, at least because the variance of inter-arrivals for each server is strongly modified.

If we assume that "only the flow" is changed, the new pdf of arrivals is $A_{n}\left(t\right)=n\, a\left(n\, t\right)$. The global shape parameters are kept, and among them the squared variation coefficient ($svc$) of the global arrivals. Let $S_{1}\left(z\right)$, $S_{n}\left(z\right)$ and $S_{r}\left(z\right)$ be the mgf of respectively $a$, $a_{n}$ and the arrival process in a single queue. When using rand, some elementary algebra leads to :

$$S_{1}\!\left(z\right)\!\!=\!1+\frac{z}{\lambda}+\frac{z^{2}}{2\la... ...z\right)^{2}}+z^{3}O\!\left(\!\negmedspace\frac{1}{n^{2}}\negmedspace\!\right)}$$

showing that random split flows are reshaped towards Poisson distribution. Especially, $svc\left(a_{r}\right)=\left(n-1+svc\left(a\right)\right)/n$ : when $svc=svc\left(a\right)>1$ the variance of arrivals in a single queue is lower than the variance of arrivals in an isolated queue.

Table 2: Dependency of sojourn time on the pooling factor

			idealized	exhaustive policies			non exhaustive policies
	$n$		fast	load	jsiz	jran	size	rtwo	robn	rand
	7		339.1	458.6	481.2	550.6	497.61	627.2	921.7	2070.0
	5		470.5	586.7	606.5	675.5	621.5	710.7	997.2	2102.7
	3		794.5	881.2	898.7	965.2	911.7	948.0	1242.9	2141.7
	1		2378.2	2378.2	2378.2	2378.2	2378.2	2378.2	2378.2	2378.2

The fact that greater $n$ leads to better performances is even more clear for the remaining policies. When using robn, $svc\left(a_{r}\right)=svc\left(a\right)/n$ holds : at the limit, only the $svc$ of the services will contribute to the waiting process. When using fast, the distribution of the number of customers staying in the system is independent of the flow itself, but only depends on the shape parameters. Therefore, according to Little's formula, the mean sojourn is divided by $n$.

Table 3: Five servers, increasing load

	$\rho$	fast	load	jsiz	size	jran	rtwo	robn	rand
	0.933	470.5	586.7	606.5	621.5	675.5	710.7	997.2	2102.7
	0.973	1139.3	1220.2	1243.9	1309.8	1333.8	1346.7	2348.4	4983.1
	ratio	2.42	2.08	2.06	2.11	1.97	1.89	2.35	2.37