Average Queue Length

Once again, let's consider a system with a buffer and a processor that are serially connected as shown in Figure 4.1. To avoid collisions of multiple inputs at the processor, the buffer stores inputs while the processor is busy working on previous inputs.

Depending on inter-arrival times of between inputs and Processor's processing time, the length of time an input waits in Buffer can vary widely. Thus the number of waiting inputs (queue size) can be a random number.

Recall how we developed the probability that the current state $C$ is equal to a state $s$ in Section 4.1.3. Let the current state $C$ of Buffer be defined as the number of inputs currently waiting in buffer. Then the probability that the number of waiting parts $C$ is equal to $x \in \mathbb{N}$ , where $\mathbb{N}$ is a suitably defined subset of the natural numbers, over an observation time from 0 to $t_o$ is

$$P(C=x)=\frac{\underset{i \in \mathbb{N}}{\sum}td(x,i)}{t_o}$$ (5.6)

The mean or expected value of $C$ is defined by

$$E(C)=\underset{x \in \mathbb{N}}{\sum} x P(C=x)$$ (5.7)

The Average Queue Length is defined as Equation (4.7).

Example 5.4   Suppose that we have a state trajectory of a queue as shown in Figure 4.4. By Equation (4.6), we can get $P(C$ =0)=(4+7)/60=0.183, $P(C$ =1)= (3+3+3+5+7)/60=0.35, $P(C$ =2)=(4+5+7+3)/60=0.317, $P(C$ =3)=9/60=0.15. By Equation (4.7), the Average Queue Length is $E(C=x)$ =0*0.183+1*0.35+ 2*0.317+3*0.15=1.434. $\square$

Figure 4.4: Trajectory of Queue

Since the natural number $x \in \mathbb{N}$ is the special case of a general state $s \in S$ , if we can calculate $P(C=s)$ then we can also calculate $P(C=x)$ as well as $E(C)$ . We will see how we implement this process in Section 4.2.3.