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.

- 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 .] - 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).] - 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.

Mon Feb 26 19:16:06 PST 1996