# Queuing theory

• Oct 10th 2009, 01:37 AM
reenasingh
Queuing theory
Hi,

I am not able to solve thisquestion satisfactorily..it would be great if anyone can suggest it ssolution.

A restaurant has N tables, with a customer seated at each table. Two waiters are
serving them.One of the waiters moves from table to table taking orders for food. The
time that he spends taking orders at each table is exponentially distributed with parameter
μ1 . He is followed by the wine waiter who spends an exponentially distributed time with
parameter μ2 taking orders at each table. Customers always order food first and then
wine, and orders cannot be taken concurrently by both waiters from the same customer.
All times taken to order are independent of each other. A customer, after having placed
her two orders, completes her meal at rate ν, independently of the other customers. As
soon as a customer finishes her meal, she departs and a new customer takes her place
and waits to order. Model this as a closed migration process. Show that the stationary
probability that both waiters are busy can be written in the form
{G(N − 2)/ G(N )} *(vsquare/μ1* μ2)
for a function G(·), which may also depend on ν, μ1 , μ2 , to be determined.
In the above model it is assumed that the restaurant is always full. Develop a model in
which this assumption is relaxed: for example, assume that customers enter the restaurant
at rate λ while there are tables empty. Again obtain an expression for the probability that
both waiters are busy.

Thank you.
Reena.