7.4.1 Average Number of Customers in the System (L)

Consider a queuing system over a period of time (T) and let (L(t)) denote the number of customers in the system at time (t). Let (T_i) be the total time in ([0,T]) in which the system contained exactly (i) customers. In general (sum_{i=0}^{infty}T_i = T). The average number of customers in the system is estimated by [ hat{L}=frac{1}{T}sum_{i=0}^infty iT_i=sum_{i=0}^infty i frac{T_i}{T}. ] Notice that (T_i/T) is the proportion of time the system contains exactly (i) customers.

Let’s consider an example. Figure 7.1 gives a simulation of a queue in an interval of 20 time units. It can be seen that (T_0= 3), (T_1 = 11), (T_2 = 5) and (T_3 =1), and therefore (hat{L}= (0cdot 3 + 1cdot 11 + 2cdot 5 + 3cdot 1)/20 = 24/20 = 1.2) customers.

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By looking at Figure 7.1 it can be seen that the total area under the function (L(t)) can be decomposed into rectangles of length (T_i) and height (i), thus having area (iT_i). It follows that the total area is given by [ sum_{i=0}^{infty}iT_i = int_0^TL(t)dt ] and therefore [ hat{L}= frac{1}{T}sum_{i=0}^{infty}iT_i=frac{1}{T}int_{0}^TL(t)dt ]

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Many queuing systems exhibit some kind of long-run stability in terms of their average performance. For such systems, as time (T) gets large, the observed average number of customers in the system (hat{L}) approaches a limiting value, say (L), which is called the long-run average number in system. With probability 1 we have that [ hat{L}=frac{1}{T}int_{0}^TL(t)dt rightarrow L mbox{ as } T rightarrow infty ] If a simulation run length (T) is sufficiently long, the estimator (hat{L}) becomes arbitrarily close to (L).

The above equations can be applied to any subsystem of a queuing system. If (L_Q(t)) denotes the number of customers waiting in queue, and (T_i^Q) denotes the total time in ([0,T]) in which exactly (i) customers are waiting in queue, then [ hat{L}_Q=frac{1}{T}sum_{i=0}^{infty}iT_i^Q=int_0^TL_Q(t)dt rightarrow L_Q mbox{ as } T rightarrow infty ]

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