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EEL6507 Spring 2009, Lecture 02, Friday 2008/01/09 (Notes created by Nathan Blythe)

Defined queue variables and functions. Derived Little's result.

Queuing variables and functions

$C n$ . The $n$ ^th customer in the system.
$N (t)$ . The number of customers in the system at time $t$ .
$U (t)$ . The amount of unfinished work in the system at time $t$ , measured in units of time (such as seconds). $U (t) > 0$ implies that the system is busy; $U (t) = 0$ implies that the system is idle.
$τ n$ . The arrival time of customer $C n$ . This is an absolute time. The numerical value of $τ n$ is derived from the arrival pattern distribution (a random variable).
$X n$ . The service time for customer $C n$ . This is a time interval. The numerical value of $X n$ is a sample of the service pattern distribution (a random variable).
$W n$ . The waiting time for customer $C n$ . This is a time interval. $W n$ is a function of the state of the queue (and thus all prior customers).
$S n$ . The system time for customer $C n$ . $S n = W n + X n$ . This is a time interval.
$α(t)$ . Total number of customer arrivals on the interval $(0, t)$ .
$δ(t)$ . Total number of customer departures on the interval $(0, t)$ . I.e. total number of customer's serviced at time $t$ .

Note that $N (t) = α(t) - δ(t)$ .

These variables and functions can be plotted to give a visual interpretation of how they interact. For examples, see figures 2.2 and 2.3 in the textbook. In brief, $U (t)$ is a sum of sawtooth curves and $α(t)$ and $δ(t)$ are staircase curves. The location and sizes of the steps and peaks relate to the variables that specify the customer behavior.

Deriving Little's result

First we define

$\lambda_t = \frac{\alpha(t)}{t}$ .

$λ t$ is a measure of the average arrival rate up to time $t$ ; this is an empirical average, not the true mean of the arrival pattern distribution. Next we define

$\gamma(t) = \int_{0}^{t} N(\tau) d\tau \leq \sum_n S_n$ .

$γ(t)$ is the total system time of all customers in the system on the interval $(0, t)$ . For insight into this equation, consider the derivative $d γ(t) / d t = N (t)$ ; the instantaneous change in total accumulated system time is the number of customers in the system at that time.

Now we ask "What is the average system time per customer on some interval $(0, t)$ ?". We define

$T_t = \frac{\gamma(t)}{\alpha(t)}$ .

This is the total accumulated system time at time $t$ divided by the total number of customers that have arrived by that same time. Next we ask "How many customers are in the system, on average, over the interval $(0, t)$ ?". We define

$N_t = \frac{\gamma(t)}{t}$ .

Referring to the definition of $γ(t)$ , we see that our definition of $N t$ is equivalent to

$N_t = \frac{\int_{0}^{t} N(\tau) d\tau}{t}$

which is just the average of the number of customers in the system at time $t$ by a simple continuous average.

Rearranging and combining the definitions of $T t$ and $N t$ :

$\begin{align} \alpha(t) T_t &= \gamma(t) \\ t N_t &= \gamma(t) \\ t N_t &= \alpha(t) T_t \end{align}$

Adding the definition of $λ t$ to the mix:

$\begin{align} \alpha(t) &= t \lambda_t \\ t N_t &= t \lambda_t T_t \\ N_t &= \lambda_t T_t \end{align}$

Now, assuming that both the average arrival rate and the average system time converge, as in

$\begin{align} \lim_{t \to \infin} \lambda_t &= \lambda \\ \lim_{t \to \infin} T_t &= \bar{T}, \end{align}$

we can state Little's result.

$\begin{align} \lim_{t \to \infin} N_t &= \lim_{t \to \infin} \lambda_t T_t \\ \bar{N} &= \lambda \bar{T} \end{align}$

Little's result states that in the mean (assuming steady-state, converging behavior and all that hand-waving) the number of customers in the system is equal to the rate of customer arrivals multiplied by the time a customer spends in the system. Note that the units (customers per unit time, multiplied by unit time, yielding customers) work out cleanly.

Lecture Recording

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EEL6507sp09L02

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Contents

EEL6507 Spring 2009, Lecture 02, Friday 2008/01/09 (Notes created by Nathan Blythe)

Queuing variables and functions

Deriving Little's result

Lecture Recording

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