Persons arrive at a taxi stand with room for W taxis according to a Poisson process with rate . A person boards a taxi upon arrival if one is available and otherwise waits in line. Taxis arrive at the stand according to a Poisson process with rate . An arriving taxi that finds the stand full departs immediately; otherwise, it picks up a customer if at least one is waiting, or else joins the queue of waiting taxis.

1. Use an formulation to obtain the steady-state probability distribution of the persons queue. What is the steady-state probability distribution of the taxi queue size when W = 5 and and are equal to 1 and 2 per minute, respectively?

[ Answer: Let prob. of i taxis waiting. Then, , , , , , and .]

2. In the leaky bucket flow control scheme packets arrive at a network entry point and must wait in queue to obtain a permit before entering the network. Assume that permits are generated by a Poisson process with given rate and can be stored up to a given maximum number; permits generated while the maximum number of permits is available are discarded. Assume also that packets arrive according to a Poisson process with given rate. Show how to obtain the occupancy distribution of the queue of packets waiting for permits. Show how to obtain the occupancy distribution of the queue of packets waiting for permits.

[ Hint: This is the same system as the one of part (1).]

3. Consider the flow control system of part (2) with the difference that permits are not generated according to a Poisson process but are instead generated periodically at a given rate. (This is a more realistic assumption.) Formulate the problem of finding the occupancy distribution of the packet queue as an M/D/1 problem.