# Homework Help: Queuing Theory problem

1. Mar 8, 2014

### saei

1. The problem statement, all variables and given/known data

I have a queuing problem. In an airport, 700 passengers arrive at the costume area uniformly distributed over a 20 minutes period. 65% are citizen and 35% are foreigner. we have 16 costume agents each of them handles either citizens or foreigners (not both). the times of costume agents to process citizens are exponentially distributed with 40 seconds and the service time for foreigners are exponentially distributed with mean 75 seconds. There are two queues, one for citizens and the other for foreigners. when a costume agent becomes free, the first one in appropriate queue goes to the agent. how many agents should serve citizens and home many should serve foreigners in order to minimize the mean time (time that a passenger arrive at the costume area to the time that the passenger completes service with a costume agents).

2. Relevant equations

I try to Use M/M/m equations but the traffic intensity becomes greater than 1 and system will go to un-stability.

3. The attempt at a solution

2. Mar 8, 2014

### Staff: Mentor

I think after those 20 minutes, no more passengers arrive and the agents have enough time to clear the queue.

Given the large number of passengers and the small number of possible answers (1 to 15), I wonder if it could be sufficient to consider the average cases for some unknowns.

3. Mar 9, 2014

### haruspex

Yes, that's how I read it.

Also, the arrival rate to each queue so far exceeds the mean service times that it might be a reasonable approximation to suppose that the queues do not fall empty before the 20 minutes. This means we can just consider the emptying of two queues of known initial length.

I would start by evaluating the total average load on each queue (passengers x service time). As mfb says, this will give you a good first guess at the answer.

If there are n agents serving a queue, each with exponential service time parameter λ, what is the service time distribution for the queue?

4. Mar 9, 2014

### saei

Thank you

Thank you for your help, but I am still confusing :D

5. Mar 9, 2014

### haruspex

Given the average service time of a citizen and the number of them, what is the average total agent time to serve all of them? Similarly for the foreigners. How do those times compare?

6. Mar 9, 2014

### saei

Unfortunately, there is not any other information :(

7. Mar 9, 2014

### haruspex

You have all that information.

8. Mar 9, 2014

### saei

I calculated the mean total service time for all the citizens and foreigners.
for citizens : 18200 seconds
for foreigners : 18375 second
Now, according to the fact that total number of citizens are more than foreigners, I should assign more agents for citizens in order to reduce the waiting time of them in the queue to achieve the minimum response time.
Is it true?
But, I could not calculate the response time, because all the formula that I have are for steady state systems and it is not steady state. !!!

9. Mar 10, 2014

### Staff: Mentor

That is true.

I don't know how to treat this properly without approximations.

10. Mar 12, 2014

### Ray Vickson

I agree with the other posters that the problem is to clear out all 700 passengers, with no new passengers arriving.

The total service times of citizens and foreigners are fixed and uncontrollable by any policy, so we might as well neglect them and just try to minimize the average queueing time of customers; these are the times customers spend in their queues, before starting service.
If we assign nc servers to customers, then nc of the customers spend no time in their queue, while the remaining Nc = 455-nc do spend time in the queue. The queueing time of the first customer in the queue is ETc/nc, where ETc = average service time of a citizen; this is true because the rate that citizens leave service (when all servers are busy) is nc/ETc; in fact, the time until his service is the minimum of nc iid exponential random variables, which is again exponential with rate of nc times the rate per server. Similarly, the second customer in the queue spends an average time of 2*ETC/nc waiting for service, while the third customer waits an average of 3*ETc/nc, etc. So, what is the average of all the citizens' queueing times, in termns of nc?

Do the same computation for the foreigners, where nf servers are assigned to them. Now figure out how to combine the averages of the citizens and the foreigners into some type of overall average; this will be a not-too-complicated function of nc and nf. You want to minimze it, subject to having nc+nf = 16.

11. Mar 13, 2014

### haruspex

So let's try a fairly crude approach.
If there are n in a queue, and the expected time to service the whole queue is t, what is the expected total waiting time?