Birth Death Processes

TW70

9N

ET20

12

Lect

ure

12.2

Birth Death Processes Ashour

0

0λ

1

1λ

2

2λ

k-1

1kλ −

k

kλ

K+1

1kλ +

. . . .

ng

rsity

in C

airo

States of the process may represent a

1μ 2μ 3μ kμ

k

1kμ +

K+1

2kμ +. . . .

ce M

odel

in G

erm

an U

nive

r States of the process may represent a count of something ( number of packet in a queue The population of a city the

erfo

rman

coh

amed

Ash

our, a queue, The population of a city, the

number of customers in store)

P Mo

W70

9Le

ctur

eN

ETW

2012

2

2.2

Birth Death Processes Ashour

0

0λ

1

1λ

2

2λ

k-1

1kλ −

k

kλ

K+1

1kλ +

. . . .

ng

rsity

in C

airo

1μ 2μ 3μ kμ

k

1kμ +

K+1

2kμ +. . . .

00 1 10 p pλ μ= +−

ce M

odel

in G

erm

an U

nive

r

1 1 11 ( )0 k k kk k k kp p pλ λ μμ− + +−= − + +

0λ

erfo

rman

coh

amed

Ash

our,

12 12 1 00( )p p pμμ λ λ= + −

01 0

1

p pμ

=

01 1 02 02 0)(p p p

λλ μμ

μλ= + −

P Mo

W70

9

1μ0 1

2 01 2

p pλ λμ μ

= 0 1 23 0

1 2 3

p pλ λ λμ μ μ

=

Lect

ure

NET

W20

12

3

2.2

0 1 2 10

1 2 3 1k

k k

k k

p pλ λ λ λ λμ μ μμ μ

−

−=

Ashour

1

0

i

ki

kp p

λμ

= −= ∏

ng

rsity

in C

airo

10 ii μ +=

1kp∞

=∑1

0 0 1i k

ip pλλ

∞ = −+ =∑ ∏ 0 1

1i k

pλ∞ = −

=

ce M

odel

in G

erm

an U

nive

r

0k

k=∑

01 1ik iλ +==

01 1

1 i

iik

λμ== +

+∑ ∏

erfo

rman

coh

amed

Ash

our,

1

10 1

1

1

i k

k i ki i

i

i

pλμλ

= −

∞ = −= +

=+

∏∑ ∏P M

oW

709

01 1

1ik iμ= +=

+∑ ∏

Lect

ure

NET

W20

12

4

2.2

Queuing SystemAshour S Servers in

the system

ng

rsity

in C

airo

k units in the queue

ce M

odel

in G

erm

an U

nive

r

kλ kμA i l Departure

erfo

rman

coh

amed

Ash

our,

mK maximum queue size Arrival

RateDeparture

Rate

P Mo

W70

9Le

ctur

eN

ETW

2012

5

2.2

Queuing notation Ashour

ng

rsity

in C

airo

M/M/S/KM i

ce M

odel

in G

erm

an U

nive

r

Arrival Process(M refers to exponential inter-arrival )

Maximum queue size

erfo

rman

coh

amed

Ash

our,

Departure Process(M refers to exponential inter-departure )

Number of Servers in the

tP Mo

W70

9

inter departure ) system

Lect

ure

NET

W20

12

6

2.2

M/M/1Ashour

0

λ

1

λ

2

λ

k-1

λ

k

λ

K+1

λ

. . . .

ng

rsity

in C

airo

Exponential inter-arrival

μ μ μ μ μ μ

ce M

odel

in G

erm

an U

nive

r Exponential inter arrival Exponential inter-departureOnly one server

erfo

rman

coh

amed

Ash

our, Only one server

Unlimited queue size

P Mo

W70

9 All arrival rates are equal All departure rates are equal

Lect

ure

NET

W20

12

All departure rates are equal

7

2.2

M/M/1 Probability of queue length k Ashour

1

1

1 i k

k i kip

λλ

= −

∞ == ∏ 1

k

k kp

λ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟

ng

rsity

in C

airo

10

01

1

1

1k i k

i

i

i

k

i

i

μλμ

∞ = −=

= +=

++∏

∑ ∏1

1

k k

k

pμλ

μ

∞

=

⎟⎜ ⎟⎝ ⎠⎛ ⎞⎟⎜+ ⎟⎜ ⎟⎟⎜⎝ ⎠∑

1

ce M

odel

in G

erm

an U

nive

r

if 211 .... 1

1x x x x

x∞= + + + <

−

k⎛ ⎞

erfo

rman

coh

amed

Ash

our,

1

11

1

k

k

λμ λ

μ

∞

=

⎛ ⎞⎟⎜+ =⎟⎜ ⎟⎜ ⎟⎝ ⎠ −∑ 1

λμ

<Queue Stability condition

P Mo

W70

9

1k

kpλλ ⎛ ⎞⎛ ⎞ ⎟⎟⎜⎜= − ⎟⎟⎜⎜ ⎟⎟⎟⎜ ⎟⎜

μ

( )( )1k

kp ρ ρ= −

Lect

ure

NET

W20

12

8

2.2

k μ μ ⎟⎟⎟⎜ ⎟⎜⎝ ⎠⎝ ⎠ ( )( )1kp ρ ρ

M/M/1 Average number of customers in the system

Ashour

customers in the system

kN kp∞

=∑

ng

rsity

in C

airo

0k

k

p=∑

(1 ) kN kρ ρ∞

= − ∑

ce M

odel

in G

erm

an U

nive

r

0k=

1(1 ) kN k dρ ρ ρ ρ∞

−⎛ ⎞⎛ ⎞∂ ⎟⎟⎜ ⎜ ⎟⎟= − ⎜ ⎜ ⎟⎟⎜ ⎜∑∫ (1 ) kN ρ ρ ρ

∞⎛ ⎞∂ ⎟⎜ ⎟= − ⎜ ⎟⎜∑

erfo

rman

coh

amed

Ash

our,

0

(1 )k

N k dρ ρ ρ ρρ =

⎟⎟⎜ ⎜ ⎟ ⎟⎜⎜∂ ⎝ ⎠⎝ ⎠∑∫

0

(1 )k

N ρ ρ ρρ =

⎟⎜ ⎟⎜∂ ⎝ ⎠∑

1(1 )N ρ ρ

⎛ ⎞∂ ⎟⎜= − ⎟⎜ ⎟1

(1 )N ρ ρ⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜P M

oW

709

(1 )1

N ρ ρρ ρ

⎟⎜ ⎟⎟⎜∂ −⎝ ⎠

Nρ

=

( )2(1 )

1N ρ ρ

ρ⎟= − ⎜ ⎟⎜ ⎟⎜ ⎟− ⎟⎜⎝ ⎠

Lect

ure

NET

W20

12

9

2.2

1N

ρ=

−

M/M/1 queue length Variance Ashour

( )22

0k

k

k N pσ∞

== −∑ ( ) ( )22

0

1 k

k

k Nσ ρ ρ∞

== − −∑

⎛ ⎞

ng

rsity

in C

airo

( ) ( )2 2 2

0 0 0

1 2k k k

k k k

k N k Nσ ρ ρ ρ ρ∞ ∞ ∞

= = =

⎛ ⎞⎟⎜ ⎟= − − +⎜ ⎟⎜ ⎟⎜⎝ ⎠∑ ∑ ∑⎛ ⎞ ( )21

ρ

ce M

odel

in G

erm

an U

nive

r

( ) ( )( )

2 2 22

0

11 2

11

k

k

k N Nρ

σ ρ ρρρ

∞

=

⎛ ⎞⎟⎜ ⎟⎜ ⎟= − ⎜ − + ⎟⎜ ⎟−⎜ ⎟− ⎟⎜⎝ ⎠∑

( )1 ρ−

erfo

rman

coh

amed

Ash

our,

( ) ( )( ) ( )2

2 2 12

0

21

1 1

k

k

k d Nρ ρ

σ ρ ρ ρ ρρ ρ ρ

∞−

=

⎛ ⎞⎛ ⎞ ⎟⎜ ⎛ ⎞ ⎟⎟⎜⎜ ∂ ⎟⎟ ⎟⎜ ⎜⎜ ⎟⎟ ⎟⎜= − − ⎜ −⎜ ⎟⎟ ⎟⎜ ⎜⎜ ⎟⎟ ⎟∂ ⎜ ⎟ ⎜⎜ ⎟⎟⎝ ⎠ ⎟⎜⎜∑∫

P Mo

W70

9

( ) ( )0 1 1kρ ρ ρ=∂ ⎟⎟⎝ ⎠ − − ⎟⎜⎜ ⎟⎝ ⎠ ⎟⎜⎝ ⎠

2ρ∞⎛ ⎞⎟⎛ ⎞⎜ ∂ ⎟⎜ ⎜

Lect

ure

NET

W20

12

10

2.2

( )( )

23

0

11

k

k

kρ

σ ρ ρ ρρ ρ=

⎛ ⎞∂ ⎟⎜ ⎟⎜ ⎟⎟= − ⎜ −⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜∂⎜ ⎝ ⎠ ⎟− ⎟⎜⎝ ⎠∑

Ashour

2ρ ρ⎛ ⎞⎛ ⎞ ⎟⎟⎜ ⎜∂ ⎟⎟⎜

ng

rsity

in C

airo

( )( ) ( )

22

2 31

1 1

ρ ρσ ρ ρ

ρ ρ ρ

⎜∂ ⎟⎟⎜ ⎜ ⎟⎟⎜= − ⎜ − ⎟⎟⎜ ⎜ ⎟⎟∂⎜ ⎜ ⎟⎟− −⎟⎜ ⎟⎜ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎟⎜

ce M

odel

in G

erm

an U

nive

r

( )( ) ( ) ( )

22

3 2 3

2 11

1 1 1

ρ ρσ ρ ρ

ρ ρ ρ

⎛ ⎞⎛ ⎞ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎜= − ⎜ + − ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜ ⎟⎟− − −⎟⎜ ⎟⎜ ⎝ ⎠⎝ ⎠

erfo

rman

coh

amed

Ash

our,

( )( ) ( )

2 22

3 31

1 1

ρ ρ ρσ ρ

ρ ρ

⎛ ⎞⎟⎜ + ⎟⎜ ⎟= − ⎜ − ⎟⎜ ⎟⎜ ⎟− − ⎟⎜⎝ ⎠P Mo

W70

9

( ) ( )1 1ρ ρ ⎟⎜⎝ ⎠

( )2

1

ρσ =

Lect

ure

NET

W20

12

11

2.2

( )1 ρ−

M/M/1 Delay Ashour

N

ng

rsity

in C

airo

NT

λ=

1 1

ce M

odel

in G

erm

an U

nive

r

11

Tρρ λ

=−

1 1

1T

λ μμ

=−

erfo

rman

coh

amed

Ash

our,

μ

1T =P M

oW

709

Tμ λ

=−

Lect

ure

NET

W20

12

12

2.2

Queue Length Survivor Function Ashour

( ) kP q k p∞

≥ =∑

ng

rsity

in C

airo

( ) kk

q p≥ ∑

( ) (1 ) iP q k ρ ρ∞

≥ = −∑ ( ) (1 ) k iP q k ρ ρ ρ∞

≥ = − ∑

ce M

odel

in G

erm

an U

nive

r

i k= 0i=

( ) kP q k ρ≥ =

erfo

rman

coh

amed

Ash

our, ( )q ρ≥

P Mo

W70

9Le

ctur

eN

ETW

2012

13

2.2

M/M/∞Ashour

0

λ

1

λ

2

λ

k-1

λ

k

λ

K+1

λ

. . . .

ng

rsity

in C

airo

I fi it b f

μ 2μ 3μ kμ ( 1)k μ+ ( 2)k μ+

ce M

odel

in G

erm

an U

nive

r Infinite number of servers Each server has departure rate

erfo

rman

coh

amed

Ash

our,

As the number of customers in the systems increases the departure rate

P Mo

W70

9

y pincreasesAny arriving customer will find an empty

Lect

ure

NET

W20

12

Any arriving customer will find an empty server

14

2.2

Queue length ProbabilityM/M/∞Ashour

1

10 1

1

1

i k

k i ki i

i

i

pλμλ

= −

∞ = −= +

= ∏∑ ∏

ng

rsity

in C

airo

0

01

1

1

1 i

i

i

k

i

i

μλμ

=

= +=

++∑ ∏

11 i k λ= −

∏

ce M

odel

in G

erm

an U

nive

r

10

01 ( 1

( 1)1

)

k i ki

ik

p

i

i μλμ

∞ = −=

== +

=+

+∏

∑ ∏k⎛ ⎞

erfo

rman

coh

amed

Ash

our,

1 1!1

1!

k

k kp

k

k

λμλ∞

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠⎛ ⎞⎟⎜+ ⎟⎜ ⎟⎟⎜∑

P Mo

W70

9

1 !k k μ=⎟⎟⎜⎝ ⎠

∑

ke

λμ λ

−⎛ ⎞

eλμ

Lect

ure

NET

W20

12

15

2.2

!ke

pk

μ λμ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

Average Queue length M/M/∞Ashour

0k

k

N kp∞

=∑

ng

rsity

in C

airo

0k=

1

0!

k

k

eN k

k

λμ λ

μ

−∞ ⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎟⎜⎝ ⎠∑ ( )

1

1

01 !

k

k

eN

k

λμλ λ

μ μ

− −∞ ⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎟⎜− ⎝ ⎠∑

ce M

odel

in G

erm

an U

nive

r 1 !k k μ= ⎝ ⎠

Nλ

=

( )1 1 !k kμ μ= − ⎝ ⎠

erfo

rman

coh

amed

Ash

our, μ

P Mo

W70

9

1T

μ=Delay

Lect

ure

NET

W20

12

16

2.2

Queue Length VarianceM/M/∞Ashour

( )22k

k

k N pσ∞

= −∑ ( )22

!

k

k

e k Nk

ρ ρσ

∞−= −∑

ng

rsity

in C

airo

0k= 0 !k k=

( )2 2 2

0

2!

k

k

e k kN Nk

ρ ρσ

∞−

== − +∑

ce M

odel

in G

erm

an U

nive

r 0k

( )1 1

2 2

1 1 0

2( 1)! !1 !

k k k

k k k

e k N Nk kk

ρ ρ ρ ρσ ρ ρ

∞ ∞ ∞− −−

= = == − +

−−∑ ∑ ∑

erfo

rman

coh

amed

Ash

our,

( ) ( ) ( )1 1 1

2 2

2 1 1 0

1 2( 1)! !1 ! 1 !

k k k k

k k k k

e k N Nk kk k

ρ ρ ρ ρ ρσ ρ ρ ρ

∞ ∞ ∞ ∞− − −−

= = = =

⎛ ⎞⎟⎜ ⎟⎜= − + − + ⎟⎜ ⎟−⎜ − − ⎟⎝ ⎠∑ ∑ ∑ ∑⎛ ⎞P M

oW

709

⎝ ⎠

( ) ( )2 1 1

2 2 2

2 1 1 0

2( 1)! !2 ! 1 !

k k k k

k k k k

e N Nk kk k

ρ ρ ρ ρ ρσ ρ ρ ρ

∞ ∞ ∞ ∞− − −−

= = = =

⎛ ⎞⎟⎜ ⎟⎜= + − + ⎟⎜ ⎟−⎜ − − ⎟⎝ ⎠∑ ∑ ∑ ∑

Lect

ure

NET

W20

12

17

2.2

( )2 2 2 22σ ρ ρ ρ ρ ρ= + − + =

M/M/mAshour

0

λ

1

λ

2

λ

k-1

λ

m-1

λ

m

λ

. . . .

ng

rsity

in C

airo

Departure rate will increase with the

μ 2μ 3μ (1 )m μ− mμ mμ

ce M

odel

in G

erm

an U

nive

r Departure rate will increase with the number of customers in the system until all the servers are busy

erfo

rman

coh

amed

Ash

our, all the servers are busy

When all the servers are busy the departure rate will remain the sameP M

oW

709

departure rate will remain the same

Lect

ure

NET

W20

12

18

2.2

Ashour

0

! /

k

kip

pk

λ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟k m≤

ng

rsity

in C

airo

1

10 1

1

1

i k

k i ki i

i

i

pλμλ

= −

∞ = −= +

=+

∏∑ ∏

! /kp k mμ ⎟⎜ ⎟⎝ ⎠

0k

ip λ⎛ ⎞⎟⎜

ce M

odel

in G

erm

an U

nive

r

01 1

1ik iμ= +=

+∑ ∏

( ) ( )1

1 1k k

m m mρ ρ−

− ∞⎡ ⎤⎢ ⎥

0

!m

kip

p mm

λμ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎝ ⎠

k m>

erfo

rman

coh

amed

Ash

our, ( ) ( )

01

11

! ! k mk k m

m mp

k m m

ρ ρ−

= =

⎢ ⎥= + +⎢ ⎥

⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑

1k k

−⎡ ⎤ 1−⎡ ⎤P Mo

W70

9

( ) ( ) ( )1

01

1! !

k m k mm

k mk k m

m m mp

k m m

ρ ρ ρ−

− ∞

−= =

⎡ ⎤⎢ ⎥

= + +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑ ( ) ( )1

1

1

11

! ! 1

k mm

k

m m

k m

ρ ρ

ρ

−−

=

⎡ ⎤⎢ ⎥

= + +⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

∑

Lect

ure

NET

W20

12

19

2.2

⎣ ⎦ ⎢ ⎥⎣ ⎦

M/M/1/K Finite Storage queue Ashour

0

λ

1

λ

2

λ

k-1

λ

k. . . .

ng

rsity

in C

airo

Number of states are limited to kμ μ μ μ

ce M

odel

in G

erm

an U

nive

r

Limited queue size , limited Hard Disk space, or limited number of seats in bus or cinema

erfo

rman

coh

amed

Ash

our, Same as M/M/1 queue except for

determining p0

P Mo

W70

9Le

ctur

eN

ETW

2012

20

2.2

Ashour

( )0k

p ρ⎧⎪⎪⎪⎪

k K≤

ng

rsity

in C

airo

( )0

0k

pp

ρ⎪⎪=⎨⎪⎪⎪⎪⎩k K>

ce M

odel

in G

erm

an U

nive

r

1

0 1K

kp ρ−⎡ ⎤

⎢ ⎥= +⎢ ⎥⎣ ⎦

∑1

1 k kρ ρ−∞ ∞⎡ ⎤

⎢ ⎥= + −⎢ ⎥⎣ ⎦

∑ ∑1

1 k K k Kρ ρ ρ−∞ ∞

−⎡ ⎤⎢ ⎥= + −⎢ ⎥⎣ ⎦

∑ ∑

erfo

rman

coh

amed

Ash

our, 1k=

⎢ ⎥⎣ ⎦ 1 1k k K= = +

⎢ ⎥⎣ ⎦ 1 1k k K= = +

⎢ ⎥⎣ ⎦

1

1 Kρ ρρ

−⎡ ⎤⎢ ⎥= + −⎢ ⎥

1(1 )

1Kρ ρ

−⎡ ⎤+⎢ ⎥= +⎢ ⎥1 ρ⎡ ⎤−⎢ ⎥= ⎢ ⎥P M

oW

709

1 1ρ

ρ ρ+

⎢ ⎥− −⎣ ⎦1

1 ρ+⎢ ⎥−⎣ ⎦ 1 Kρ⎢ ⎥+⎣ ⎦

1 kpρ

ρ⎡ ⎤−⎢ ⎥=

Lect

ure

NET

W20

12

21

2.2

1k

kK

p ρρ

⎢ ⎥= ⎢ ⎥+⎣ ⎦

M/M/m/m Finite Storage queue m servers

Ashour

servers 0

λ

1

λ

2

λ

m-1

λ

m. . . .

ng

rsity

in C

airo

Number of states are limited to mLimited queue size limited Hard Disk space or limited number

μ 2μ 3μ mμ

ce M

odel

in G

erm

an U

nive

r Limited queue size , limited Hard Disk space, or limited number of seats in bus or cinema

Number of servers limited to m

erfo

rman

coh

amed

Ash

our, Used to simulate a system where there is no waiting

The probability that all servers are busy is the probability that users will be blocked from the systemP M

oW

709

probability that users will be blocked from the system

Lect

ure

NET

W20

12

22

2.2

Ashour

0

!ik

kp

pk

λ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜ ⎟k m≤

ng

rsity

in C

airo

1

10 1

1

1

i k

k i ki i

i

i

pλμλ

= −

∞ = −= +

=+

∏∑ ∏

!kp k μ ⎟⎜ ⎟⎝ ⎠

ce M

odel

in G

erm

an U

nive

r

01 1

1ik iμ= +=

+∑ ∏0kp = k m>

1

1m kλ

−⎡ ⎤⎛ ⎞⎢ ⎥

erfo

rman

coh

amed

Ash

our,

1k

λ⎛ ⎞

00

1!

m

k

ipk

λμ=

⎛ ⎞⎢ ⎥⎟⎜= ⎟⎜⎢ ⎥⎟⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦∑

mP Mo

W70

9

1!

1k ik

kp

λμ

λ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠=

⎛ ⎞ 0

1!!

m

m mi

i

p

ki

ρ

ρ=

=

∑

Lect

ure

NET

W20

12

23

2.2 0

1!i iλμ=

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠∑

0i=

M/M/∞/ /M Customer populationAshour

0

Mλ

1

( 1)M λ−

2

( 2)M λ−

M-1

λ

M. . . .

ng

rsity

in C

airo

Population is limited to MAs more users arrive the rate of arrivals decrease

μ 2μ 3μ Mμ

ce M

odel

in G

erm

an U

nive

r As more users arrive the rate of arrivals decrease As more of users arrive the rate of service increases

erfo

rman

coh

amed

Ash

our,

P Mo

W70

9Le

ctur

eN

ETW

2012

24

2.2

Ashour

( ) 0k

M k k Mλ

λ⎧⎪ − ≤ ≤⎪⎪=⎨

ng

rsity

in C

airo

0k

M k⎨⎪ <⎪⎪⎩

1 ( )k M ip p

λ− −= ∏

k

p pMλ ⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟=

ce M

odel

in G

erm

an U

nive

r

( )00 1k

i

pi

pμ= +

= ∏ 0kp kp

μ⎜⎟⎜ ⎟⎜⎟⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

=⎟⎟

1kM Mλ

−⎛ ⎞⎛ ⎞⎛ ⎞⎟⎜ 1

erfo

rman

coh

amed

Ash

our,

00k

M Mp

kλμ=

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎟⎜⎝

⎟⎜ ⎟⎜⎟⎟⎜ ⎟⎟⎜⎝ ⎠ ⎠⎝ ⎠∑ 0

1

(1 / )Mp

λ μ+=

k Mλ ⎛ ⎞⎛ ⎞ ⎟⎜P Mo

W70

9

(1 / )k M

M

kp

λμ

λ μ

⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟⎜⎟⎟⎜ ⎟⎟⎜⎝ ⎠ ⎝ ⎠+

=

Lect

ure

NET

W20

12

25

2.2

( / )μ+

Average number of customers in the systems

Ashour

systems k M

kλ ⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟⎜⎟⎟⎜ ⎟⎟⎜

ng

rsity

in C

airo

(1 / )MN

kk

μ

λ μ

⎟⎟⎜ ⎟⎟⎜⎝ ⎠ ⎝ ⎠+

=∑

ce M

odel

in G

erm

an U

nive

r

0

1

(1 / )

Mk

Mk

N kM

kρ

λ μ =

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠=

+∑

erfo

rman

coh

amed

Ash

our,

0

1

(1 / )

Mk

Mk

M

kN ρ ρ

ρλ μ =

⎛ ⎞∂ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟∂ ⎜+ ⎝=

⎠∑

1 (1 / )

(1 / )

M

MN

λ μρ

ρλ μ∂ +

+=

∂

P Mo

W70

9

0( / ) kρμ ⎝ ⎠ ( / )μ

1 /M

Nρ

λ=

Lect

ure

NET

W20

12

26

2.2

1 /λ μ+

M/M/m/K/MAshour

0

Mλ

1

( 1)M λ−

2

( 2)M λ−

m 1

( )2M m λ− +

. . .( )M m λ− ( )M K λ+ ( 1)M K λ− +

( 1)M m λ− +

ng

rsity

in C

airo

0μ

1

2μ

2

3μ

m-1

mμ

. . . mmμ mμ

K-1

mμ

K. . . ( 1)m μ−

1 ( )k M ip p

λ− −= ∏ 0 k 1

k Mλ ⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟ ≤ ≤⎜

ce M

odel

in G

erm

an U

nive

r

( )00 1k

i

pi

pμ= +

= ∏ 0 k m-10kp kp

μ⎟⎜ ⎟ ≤ ≤⎜⎟⎜ ⎟⎜⎟⎟⎜ ⎟⎟⎜⎝ ⎠ ⎝ ⎠

=

1 1( ) ( )m kM i M iλ λ− −− −∏ ∏ !

k M kλ ⎛ ⎞⎛ ⎞ ⎟⎜

erfo

rman

coh

amed

Ash

our,

( )00

( ) ( )

1ki i m

M i M i

ipp

mλ λ

μμ= ==

+∏ ∏ m m k K0

!!m k

k

M kk m

ppλμ

−⎛ ⎞ ⎟⎜⎟⎜ ⎟ ≤ ≤⎜⎟⎜ ⎟⎜⎟⎟⎜ ⎟⎟⎜⎝ ⎠ ⎝ ⎠

=

k Mλ ⎛ ⎞⎛ ⎞ ⎟⎜P Mo

W70

9

imk

M

kp

M

λμ

λ

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟⎜⎟⎟⎜ ⎟⎟⎜⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞ ⎟⎜

=

Lect

ure

NET

W20

12

27

2.2 0i

M

iλμ=

⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟⎜⎟⎟⎜ ⎟⎟⎜⎝ ⎠ ⎝ ⎠∑