M/M/1 Queues

• Customers arrive according to a Poisson process with rate .

• There is only one server.

• Service time is exponential with rate .

0j λ,λ j 1j μ,μ

0μ

j

0

0 1 2 j-1 j j+1

...

M/M/1 Queues

• We let = /, so • From Balance equations:

• As the stationary probabilities must sum to 1, therefore:

0

02

201 , , ,

j

j

) 1(1

12

0

210

j

j

j

j

jjc

21

110

M/M/1 Queues• But for <1,

• Therefore:

1/1 1 2 j

1 ),1(

)1(0

jjj

M/M/1 Queues• L is the expected number of entities in the system.

1)

1()1(

)1()1(

)1(

system)in entities j(

11

1

00

0

d

d

d

dj

jj

PjL

j

j

j

j

j

j

j j

j

M/M/1 Queues

• Lq is the expected number of entities in the queue.

LL

j

j

PjL

j jj j

j j

jq

)1(

)1(

system)in entities j()1(

0

11

1

1

)(1

22

qL

Little’s Formula• W is the expected waiting time in the system

this includes the time in the line and the time in service.

• Wq is the expected waiting time in the queue.

• W and Wq can be found using the Little’s formula. (explain for the deterministic case)

qq WL

WL

1LW

M/M/1 queuing model Summary

2

1

1

qq

q

WL

WW

WL

W

1

)(

1)0(0

n

n nNP

NP

M/M/s Queues

• There are s servers.• Customers arrive according to a Poisson process

with rate ,• Service time for each entity is exponential with

rate .

• Let = /s

0 , jj

sjs

sjj

j

j

,

,

00

M/M/s Queues

• Thus

1

0

0

)1(!)(

!)(

1s

j

sj

ss

js

0 1 2 s-1 s s+1

..

. j-1 j j+1...

2 s ss s

)1( s

M/M/s Queues

2,...s1,s s,j ,!

)(

sj ,!

)(

0

0

sj

j

j

j

j

ss

s

j

s

M/M/s Queues

• All servers are busy with probability

• This probability is used to find L,Lq, W, Wq

• The following table gives values of this probabilities for various values of and s

0)1(!

)()(

s

ssjP

s

M/M/s Queues

1)(1

)(

1

)(

s

sjPLLW

LL

s

sjPLW

sjPL

q

q

qq

q

M/M/s queuing systemNeeded for steady state

• Steady state occurs only if the arrival rate is less than the maximum service rate of the system– Equivalent to traffic intensity = /s < 1

• Maximum service rate of the system is number of servers times service rate per server

M/M/1/c Queues• Customers arrive according to a Poisson

process with rate .• The system has a finite capacity of c

customers including the one in service.• There is only one server.• Service times are exponential with rate .

M/M/1/c Queues• The arrival rate is

cj

cj

j

j

,0

1 ,

1 ,

00

jj

cj ,0

cj ,

1

1

0

10

j

jj

c

M/M/1/c Queues

• L is the expected number of entities in the system.

)1)(1(

))1(1(1

1

c

cc ccL

11

1

0

1

)1(

)1)(1(

))1(1(

)1(

cc

cc

q

cc

LL

M/M/1/c Queues• We shall use Little’s formula to find W

and Wq. Note that:– Recall that was the arrival rate.– But if there are c entities in the system,

any arrivals find the system full, cannot “arrive”.

– So of the arrivals per time unit, some proportion are turned away.

c is the probability of the system being full.

– So (1- c) is the actual rate of arrivals.

M/M/1/c Queues

)1(

)1(

c

qq

c

LW

LW