Consider the M/M/1 queue. Arrival process: Poi( t) Service distribution: Poi(t) One server, infinite queue length possible Prob(arrival in small interval h ) = h Prob(service completed in small interval h ) = h We know P 0 , but what else can we say, e.g. P 1 , P 2 etc.
Possible states (sizes) of a queue
P1 = (/ ) P0 = P0 = (1-)
P2 = (+ ) P1 -P0
P2 = (1 + ) (1-) - (1-) = 2 (1-)
=0.8, N =4
=0.9, N =9
=0.95, N =17
=0.90, N =99
Q: Are results affected by queue disciplines ?
A: Not as long as queue discipline does not make explicit use of job lengths, i.e.
N = / (1-) for M/M/1 FCFS, LCFS, round robin, least attained service first
but N / (1-) for SJF
(a) What is the probability of the buffer being empty?
(b) What is the average no. of messages in the system?
(c) What is the average number of bits in the buffer?
(d) What is the average delay of a message?