Question

During lunch hour, customers arrive at a fast-food restaurant at the rate of 120 customers per hour. The restaurant has one line, with three workers taking food orders at independent service stations. Each worker takes an exponentially distributed amount of time–on average 1 minute–to service a customer. Let Xt denote the number of customers in the restaurant (in line and being serviced) at time t. The process (Xt)t≥0 is a continuous-time Markov chain. Exhibit the generator matrix.

Answer 1

This can be solved as a birth-death process.

We should either look at this as per hour or per minute. I choose to look at it as per minute.

Thus, customers arrive at a rate of 2 every minute.

The transition rates q_ij for i < j are these "births" that we see as 2 / min.

So q₀₁ = 2 which is the rate at which customers arrive. We have no serviced anyone.

q₁₀ = 1 because there is one person in line and having three cashiers won't speed up or slow down the checkout of the customer.

Meanwhile, q₁₂ = 2 which is still the "birth" rate.

As usual, Q_ii = −q_i

I.e. the diagonals are the negative holding time parameters.

We can keep going. q21 = 2 because with two customers in line we can process them at a rate of 2 customers per minute.

If we continue with this pattern we'll get up to a maximum of 3 customers per minute.