Queues forming during the Rush Hour


This example discusses how it is possible to model the build up of traffic in the rush hour, and thus to see when queues form on a straight road, without junctions, relative to the arrival rate of cars on the section of road.

Symbols Used

Symbol Represents
Q(t) Queue length at time t
A(t) Cumulative arrivals to the queue at time t
D(t) Cumulative departures from the queue at time t
mu The maximum service rate
mu(t) The service rate at time t
lambda(t) Arrival rate at time t

Modelling the Queue

In the rush hour the arrival rate, lambda(t), will rise until it reaches a maximum rate (i.e. the time when the most people reach the point of road, when they are trying to get to work), after this point the arrival rate will decrease.  There are four main points in this flow, as follows:
Time label Significance
t0 The first point at which lambda(t) equals the maximum service rate, mu
t1 The point where lambda(t) is a maximum
t2 The point where lambda(t) has decreased back down to mu
t3 The point where the queue has gone, and so mu(t) = lambda(t) 

The above details tend to imply that a graph of lambda(t) is quadratic, and so would be differentiable twice at or near t1.  This means that lambda(t) has a taylor expansion about the point t1:

Taylor expansion for lamda(t)
for constant Beta:
Equation for Beta
For small t-t1, i.e. this is valid where t0 < t < t3
This means that t0 and t2 can be estimated, by substitution:

Equations for finding t0 and t2

The length of the queue is the area between the curves of lambda(t) and mu(t), i.e.:

Finding Q(t)

By the using similar substitutions to those used in the previous derivation we can simplify this to:

Qt Final

Henry Morgan, 14 June 1999