M/M/1 Queuing

M/M/1 Queuing A queuing model that assumes one departure channel and exponentially distributed departure times in addition to exponentially distributed arrival times (an M/M/1 queue) is applicable in some traffic applications. For example, exponentially distributed departure patterns might be a reasonable assumption at a toll booth, where some arriving drivers have the correct toll and can be processed quickly, and others do not have the correct toll, producing a distribution of departures about some mean departure rate. Under standard M/M/1 assumptions, it can be shown that the following queuing performance equations apply (again assuming that ρ is less than 1):

21 Q

ρ ρ= −(5.31)( )w λμ μ λ =− (5.32)

t μ λ= −(5.33)

where

Q = average length of queue in vehicles,

w = average waiting time in the queue, in unit time per vehicle, t = average time spent in the system (w + 1/μ), in unit time per vehicle, and

other terms are as defined previously.

EXAMPLE M/M/1 QUEUING: PARKING-LOT APPLICATION

Assume that the park attendant in Examples 5.7 and 5.11 takes an average of 15 seconds to distribute brochures, but the distribution time varies depending on whether park patrons have questions relating to park operating policies. Given an average arrival rate of 180 veh/h as in Example 5.11, compute the average length of queue (in vehicles), average waiting time in the queue, and average time spent in the system, assuming M/M/1 queuing.

SOLUTION

Using the average arrival rate, departure rate, and traffic intensity as determined in Example 5.11, the average length of queue is (from Eq. 5.31)

20.75 1 0.75 2.25 veh

Q

= −=

the average waiting time in the queue is (from Eq. 5.32)

( ) 3

4 4 3 0.75 min/veh

w = −=

and the average time spent in the system is (from Eq. 5.33)

1 4 31 min/veh

t = −=