## 4.3 Dynamic and Steady State Queuing

As noted in Section 4.1, the key characteristics that determine the performance of a queuing system are the average arrival rate, r, the average service rate, s, and their ratio,\(\rho =r/s\) , known as the utilisation factor (or, sometimes, the **intensity**) of the queuing system.

Typically, instantaneous arrival and service rates vary probabilistically from one moment to the next, but each of the **average** rates of arrival and service may be considered either to remain constant or to vary in some way, over an analysis period.

An example of variation of average rates is that over a two-hour peak period, the arrival rate of vehicles on a minor road at an unsignalised intersection is likely to increase from a low, off-peak level, build to a maximum peak rate, then steadily decrease toward the end of the peak. Over the same period, a similar pattern of growth and decay in the major road traffic would influence the number of acceptable gaps for minor road vehicles to enter the intersection, which, in turn, would first decrease the intersection’s effective service rate for minor traffic to a minimum value, then steadily increase it again towards the end of the peak.

The **behaviour** of a queuing system, as reflected in queue lengths and delays, may vary over an analysis period for one of two reasons:

- Over the period, there is variation in the average arrival rate and/or the average service rate.
- Both rates remain constant over the analysis period but the average arrival rate equals or exceeds the average service rate (i.e. the utilisation factor, ρ, is greater than or equal to one).

Case (1) is what is normally termed **dynamic** queuing and applies to situations such as the growth and decay of traffic over a peak period, an example of which was discussed earlier in this subsection. In such situations, the analyst may be interested in the maximum queue lengths and delays that occur, given that vehicle arrival rates may approach or even exceed service rates over different parts of the peak.

Analysis of a dynamic queuing situation may assume deterministic or probabilistic behaviour but, in either case, the analysis period is normally divided into a succession of time intervals. Over each interval, the average arrival rate and the average service rate (and hence also the utilisation factor) can be considered to be constant, though they may vary from interval to interval. The analysis typically assumes a state of the system (as indicated by characteristics such as queue length) at the start of the first interval, then takes the state at the start of each succeeding interval to be the state at the end of the one before it.

If the queuing behaviour can be considered to be deterministic, the analysis process is relatively straightforward – for example, Figure 4.1 might represent the deterministic analysis of one interval in the examination of a dynamic queuing situation over a longer analysis period.

If the queuing behaviour is considered probabilistic, the analysis process can be much more complex, involving the application of probability theory to the analysis of each interval and producing a distribution of possible system states (rather than a single, known state) at each point of change from one interval to the next. Fortunately, however, approximation methods, such as the coordinate transformation method of Kimber and Hollis (1979), have been developed to allow the estimation of average queue lengths and average delays without the need for complex probabilistic calculations.

Case (2) above is dynamic (time dependent), even though average arrival and service rates are constant over time, because a utilisation factor greater than or equal to one results in queue lengths and delays growing steadily with time. This situation clearly cannot be sustained over a long period but could apply during a limited interval within a peak period.

Where average arrival and service rates each remain constant over an analysis period and, in addition, the utilisation factor is less than one, the queuing situation is said to be **steady state**. This means that average queue lengths and average delays will be constant over time and, in the case of probabilistic behaviour, it will be possible to derive probability distributions, which also will not vary with time, for aspects of queuing behaviour such as queue lengths and delays.

For a wide range of traffic engineering applications, analysis of steady state queues with randomly distributed arrivals and service times provides suitable guidance for road design and traffic management decisions. Section 4.4 addresses this type of analysis.