Consider the MM1 queue

1
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 P0 , but what else can we say, e.g. P1 ,
  P2 etc

2
Possible states (sizes) of a queue
?
?
?
?
...
State
0
2
1
n-1
n
n1
?
?
?
?

• Pn is how often system is in state n ( n jobs in
  queue)
• M/M/1 queue Consider small interval of time h
• P(Jump from state n-1 to n) P(arrival) ?h
  (flow from n-1 to n)
• P(Jump from state n to n-1) P(departure) ?h
  (flow from n to n-1)
• Flow between states must be balanced. If not .
• Lets look at states 1 and 0. What is the flow
  between them ?

3
Flow equations for M/M/1 queue

• In steady state inflows outflows
• Flow into state 0 ? P1
• Flow out of 0 ?P0
• For balance ? P0 ? P1
• ? P1 (?/ ?) P0 ? P0 ? (1-?)
• Flow into state 1 ?P0 ? P2
• Flow out of 1 ?P1 ? P1
• For balance ?P0 ? P2 ?P1 ? P1

4
Steadystate probabilities

• We have P0 1- ? , P1 ? (1-?)
• ? P2 (? ?) P1 - ?P0
• ? P2 (1 ?) ? (1-?) - ? (1-?) ?2 (1-?)
• Balance equation for state n
• ?Pn-1 ? Pn1 ?Pn ? Pn
• By induction, the solution is Pn ?n (1-?)
• Check that this is a valid probability distr.
• P0 P1

5
Results for M/M/1

• P( n jobs in system) Pn ?n (1-?)
• P(at least m jobs in system)

6
Performance of M/M/1

• N E(no. in system)

• R

7
Performance depends on ?
As ??1, N ?? e.g. ?0.8, N 4 ?0.9, N
9 ?0.95, N 17 ?0.90, N 99
R
? (utilisation)
8
Bound of utilisation

• ? (?/ ?) is called the utilisation or traffic
  intensity of the queueing system
• ? determines the performance of an M/M/1 queue
• as ? ? 1, N ? ?
• So, for steady state queue, we need ? lt 1
• i.e. ?lt ?, or arrival rate is slower than service
  rate

9
Other queueing disciplines

• 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

10
Example of M/M/1

• Randomly arriving messages of variable length are
  transmitted over a channel (waiting mess. stored
  in buffer)
• Avg message length 128 octets
• Line bit rate 4800 bits per second.
• Message arrival rate 7500 per hour.
• (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?

11
FDMA for mobile phone network

• N streams ? single communication line
• stream arrival rate ? packets per second
• average transmission time for stream 1/?

12
Performance of FDMA

• What is avg. no. in system ?
• What is avg. delay ?
• How does it compare to single channel M/M/1
  (statistical multiplexing) ?