Consider an M/M/ queue with channels (servers) numbered 1, 2, On arrival, a customer will choose the lowest numbered channel that is free. Thus, we can think of all arrivals as occurring at channel 1 Those who find channel 1 busy overflow and become arrivals at channel 2 Those finding both channels 1 and 2 busy overflow channel 2 and become arrivals at channel 3, and so on

(a) What fraction of time is channel 1 busy"

(b) By considering the corresponding M/M/2 loss system, determine the fraction of time that channel 2 is busy

(c) Write down an expression for the fraction of time channel c is busy for arbitrary

c.

(d) What is the overflow rate from channel c to channel c + 1 Is the corresponding overflow process a Poisson process? A renewal process? Explain briefly

(e) If the service distribution were general rather than exponential, which (if any) of your answers to (a)-

(d) would change? Briefly ex- plain