M/M/N Queuing
M/M/N Queuing A more general formulation of the M/M/1 queue is the M/M/N queue, where N is the total number of departure channels. M/M/N queuing is a reasonable assumption at toll booths on turnpikes or at toll bridges, where there is often more than one departure channel available (more than one toll booth open). A parking lot is another example, with N being the number of parking stalls in the lot and the departure rate, μ, being the exponentially distributed times of parking duration. M/M/N queuing is also frequently encountered in non- transportation applications such as checkout lines at retail stores, security checks at airports, and so on.
The following equations describe the operational characteristics of M/M/N queuing. Note that unlike the equations for M/D/1 and M/M/1, which require that the traffic intensity, ρ, be less than 1, the following equations allow ρ to be greater than 1 but apply only when ρ/N (which is called the utilization factor) is less than 1.
( ) ρ ρρ
−== +
− 0 101
! ! 1cc
N n N
cnP
n N N
(5.34)
0 for !
nnP P n N
n ρ
= ≤ (5.35)
5.5 Queuing Theory and Traffic Flow Analysis 179
ρ−= ≥0 for !
nn n N
P P n N
N N (5.36)
( ) 10! 1
Nn NP P
N N N ρ
ρ
+> = −(5.37)
where
P0 = probability of having no vehicles in the system, Pn = probability of having n vehicles in the system,
Pn>N = probability of waiting in a queue (the probability that the number of vehicles in the system is greater than the number of departure channels),
n = number of vehicles in the system, N = number of departure channels, nc = departure channel number, and
ρ = traffic intensity (λ/μ).
