Families arrive at a taxi stand according to a Poisson process with rate . An arriving family

Families arrive at a taxi stand according to a Poisson process with rate λ. An arriving family finding N other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to a Poisson process with rate μ. A taxi finding M other taxis waiting does not wait. Derive expressions for the following quantities.

(a) The proportion of time there are no families waiting.

(b) The proportion of time there are no taxis waiting.

(c) The average amount of time that a family waits.

(d) The average amount of time that a taxi waits.

(e) The fraction of families that take taxis.

Now redo the problem if we assume that N =M =∞and that each family will only wait for an exponential time with rate α before seeking other transportation, and each taxi will only wait for an exponential time with rate β before departing without a fare.