1. Let Xn denote a Markov chain in continuous time with intensity matrix Q.
Is Q ”honest”? Find the invariant distribution for this process.
2. Let Xn be a birth death process as described in class.
(a) Show that the invariant distribution for the birth death process is
(b) Suppose that λn = 1 for all n and likewise µn = n for all n. Give an exact expression for the invariant distribution.
3. Elevator problem - In class we discussed how Markov chains can be used to determine the average amount of time someone
must wait for an elevator. Suppose that we have an elevator that can hold two people at the same time. Whenever the
elevator completes a service it then comes back to the first floor to pick up the next two customers. However, if there is only
one customer in line, the elevator then serves that customer by himself. We shall assume that the service time is exponential
at rate µ whether it is serving one or two customers. We also suppose that customers arrive at the elevator with rate λ.
Suppose that the state space is described by:
Rate process leaves
= Rate process enters
λP0′ = µP0
No one in elevator
(λ+ µ)P0 = λP0′ + µP1 + µP2
Elevator is busy, and no one is waiting
n, n ≥ 1
(λ+ µ)Pn = λPn−1 + µPn+2
n customers are waiting
(a) Draw a state diagram for this process.
(b) Assume that Pn = αnP0 is a solution to the third set of equations (where N ≥ 1). Show that α =
(c) Using the system of equations (above) and the equation P0′ +P0+
1 Pn = 1, find the invariant distribution (the Pi’s)
in terms of λ and µ. (Hint: you need to use a geometric series here.)
(d) In this system, the rate at which customers are served alone is λP0′ + µP1. What is the proportion of individuals that
ride the elevator alone (Cl)? What is Cl if λ = 1 and µ = 1?
(e) Let us define the following terms:
average number of customers in the system