Problem 5. Calls from n telephones are multiplexed on a transmission line with m circuits. Each...

Transcribed Image Text:

Problem 5. Calls from n telephones are multiplexed on a transmission line with m circuits. Each call takes up one full circuit. Each telephone alternates between an idle state and an active state. The idle time is i.i.d. exponential with mean time 1/ for all the telephones. When a telephone switches to the active state, it generates one call whose duration is i.i.d. exponential with mean 1/. A telephone goes to the idle state after (i) the call generated by the telephone finishes, or (ii) the call generated by the telephone is blocked. (a) Find the steady-state occupancy distribution of the circuits for n m. (b) For n = 5, m = 3, = = 1, find (i) P{all circuits are free}, and (ii) P{all circuits are busy}. (c) What is the blocking probability when n = m. Hint: The call arrival rate to the transmission line is dependent on the number of busy circuits. Problem 5. Calls from n telephones are multiplexed on a transmission line with m circuits. Each call takes up one full circuit. Each telephone alternates between an idle state and an active state. The idle time is i.i.d. exponential with mean time 1/ for all the telephones. When a telephone switches to the active state, it generates one call whose duration is i.i.d. exponential with mean 1/. A telephone goes to the idle state after (i) the call generated by the telephone finishes, or (ii) the call generated by the telephone is blocked. (a) Find the steady-state occupancy distribution of the circuits for n m. (b) For n = 5, m = 3, = = 1, find (i) P{all circuits are free}, and (ii) P{all circuits are busy}. (c) What is the blocking probability when n = m. Hint: The call arrival rate to the transmission line is dependent on the number of busy circuits.