# Queuing Theory with multiple class of traffics

1. May 9, 2012

### arup

I am trying to use Queuing Theory to do some computer architecture studies .

Is there any mathematical treatment of Queuing Theory in which there is single queue but multiple classes of traffic along with multiple servers such that for every class of traffic there is one dedicated server?

Is this situation equivalent to having multiple queues one for each class of traffic? Or is there more to it?

2. May 9, 2012

### chiro

Hey arup and welcome to the forums.

For this problem, is it analogous to say looking at a supermarket with multiple aisles and each aisle has its own characteristics including traffic distribution and model criteria as well as the server model that deals with the traffic?

If this is the case the answer is yes, but for clarification we need to know a bit more about the actual nature of the queue(s) itself and the server(s) themselves.

3. May 9, 2012

### arup

Let me see if I can describe an analogous situation.

1. Let's assume my Queue is a "multi-cuisine" restaurant. The restaurant has 10 tables (10 entries in the queue).
2. The restaurant offers 25 different cuisine and has one chef exclusively for each cuisine (so, 25 chefs in total).
3. The service time for each cuisine is same (say 20 mins)

Therefore, if 3 customers arrive at the empty restaurant, all wanting to eat the same cuisine (say Chinese), the 3rd customer have to wait 20 + 20 + 20 = 60 mins to get his service completed. Only one chef is working for ~60 mins while all others are idle. For a give cuisine, customers are serviced in FIFO

If however, 3 customers arrive at empty restaurant, each wanting to eat different cuisine, then each customer will get his service done in 20 mins each. 3 chefs will be working for 20 mins in parallel

So, average response time depends on the mix of traffic.

Now my question is: Can this situation be analyzed by just imagining the restaurant to be a parallel combination of 25 different queues, with the constraint that total occupancy of all queues should not be more than 10 (since there are 10 tables only)

4. May 9, 2012

### viraltux

OK, Queuing Theory appeared in a time where no computing was available and engineers needed to setup telephone lines in the most efficient way to serve phone calls demand.

I say this because Queuing theory is extremely limited for real cases and always gives you approximate results, so, if you are not involved in a purely theoretical study I strongly recommend you to dump Queuing Theory and go for a simulation all the way. If you're not familiar with programming there are a number of very easy to use general purpose simulators that will give you the right result.

Anyway, if you still want to use QT I'd say that using 25 different queues is exactly right since the servers are the cooks, not the restaurant, in this case you don't have to think in the time the costumer takes to ask for the meals since it is negligible and only think about the cooking time.

Last edited: May 9, 2012
5. May 10, 2012

### arup

Thanks Chiro and Viraltux.

I am using simulations. However just wanted to use QT to make sense of some of the results from simulation.

Even in the situation that I described above, I guess, I can still use one queue instead of 25 Qs, if I assume traffic mix remains fairly constant, such that average response time remains constant... is that correct?

6. May 10, 2012

### viraltux

Sure, you can imaging an average customer if you want and an average cook time, this is up to you and what you really want to make sense of in you simulations

7. May 10, 2012

### chiro

Given the nature of your problem, it's probably better to make sure your simulations are using the right assumptions and then to just take the expectation and variance and use these to estimate some kind of interval that corresponds to waiting time at whatever stage of the queue (starting from people lining up to finally getting served a meal) and then you can get the total waiting time.

It would probably help if you posted your simulation code so that people could critique both the model (and corresponding code) as well as the assumptions.
You will have to provide priors, especially for the people ordering a particular cuisine. If you assume a uniform prior, then use that. If however some are favored over others then your prior will not be uniform.

8. May 13, 2012

### haruspex

Not sure what you mean by the 10 tables constraint. Are you saying that there are never more than 10 customers in the entire population, or that an eleventh customer would be turned away if a total of ten already in the queues? (In the second case, I assume this is actually 'to go'. I.e. the customer spends no time eating.)