Queue Problem

[Hunter82]

I came across this problem when taking a practice exam and have no idea how to approach it. The solution seemed pretty basic but I am having a hard time figuring out how they came up with the solution equations or where they possibly came from. Any help with this problem would be greatly appreciated. The question and given solution are posted below.

Vehicles arrive at the ticket gate of a parking lot at an average rate of 30 vph. It takes an average of 1.5 min to get a ticket and drive through the gate. The arrival distribution is assumed to be Poisson and the service distribution is assumed to be exponential. There is no space limitation for the vehicles waiting to get a ticket. What is most nearly the number of vehicles expected to be waiting at the gate (i.e., the queue length)?

Q = SPV = (1veh/1.5min)(60min/hr) = 40 vph

Lq = q2/ Q(Q-q)

= (30veh/hr)2/(40veh/hr)(40 veh/hr - 30 veh/hr)

= 2.25

 

[IlPadrino]

The problem is the simplest of queues: M/M/1 (google "Kendall notation" if you've never heard of this way of designating queues)

The first M is for interarrival time distribution (memoryless aka exponential), the second M is for service time distribution (again, memoryless), and 1 is for one server.

The occupation rate (aka utilization rate or traffic rate or factor), ρ, is simply:

ρ = λ / μ, where

λ = arrival rate

μ = service rate

So... in this problem, λ = 30 vph, μ = 40 vph, and therefore, ρ = 0.75

Note that ρ has to be less than one in this sort of queue or else the queue will grow infinitely large.

Almost there... now you just need the expression for the mean number of customers in the system:

E(L) = ρ / (1-ρ) = 3 vehicles

As an aside, the expression for the mean time spent in the system:

E(S) = (1/μ) / (1-ρ)

And herein lies a nasty little distractor in this problem... you might stop at E(L) and say this is the queue length. But the problem asks how many vehicles are waiting at the gate and this implies they don't want to count the vehicle that is currently getting serviced (i.e. at the ticket dispenser). So you need the expression for the mean number of customers in the queue (not the system):

E(Lq) = E(L) - ρ = ρ2 / (1-ρ)

So... the answer is 2.25 vehicles. Makes sense?

 

[Hunter82]

Makes perfect sense now! Thanks for the detailed explanation.