primer Lecturer: Massimo Tornatore Original material prepared by: - - PowerPoint PPT Presentation

SLIDE 1

Queueing theory primer

Lecturer: Massimo Tornatore

Original material prepared by: Professor James S. Meditch Typesetter: Dr. Anpeng Huang Courtesy of: Prof. Biswanath Mukherjee

1

SLIDE 2

A change of focus

• So far we have investigated «static» problems

– Traffic requests are given and constant in time

• E.g., Multi commodity flow problem
• In general, mathematical programming, optimization and graph

theory, heuristics…

• Now we move to a class of dynamic problems

– Random or stochastic flow problems – The times at which the demands arrive are uncertain and also the size of the demands are unpredictable

• Queueing (in our case «traffic») theory

2

SLIDE 3

Source

• Notes taken mainly from

– L. Kleinrock, Queueing Systems (Vol 1: Theory)

• Chapter 1 and 2

– L. Kleinrock, Queueing Systems (Vol. 2: Computer Applications)

• Chapter 1

3

SLIDE 4

Delay and Congestion, why?

• The reason for this behaviour is the irregularity

(i.e, statistical distribution) of:

– Arrivals (i.e., interarrival times) – Services (i.e., service times)

R C

If R>C, we expect congestion (intuitive) If R<C, there might still be congestion (why?)

4

SLIDE 5

• A. Notation and terminology

5

SLIDE 6

cn = customer n = arrival time of cn tn = – = interarrival time → wn = waiting time for cn → xn = service time for cn → sn = system time for cn → sn = wn + xn

n



n

 t ~

1  n

 s ~ x ~

w ~

*Note that these distributions do not depend by n (same distribution for all arrivals/services) 6

SLIDE 7

Arrival process Service process

k

t t E t t E dt t dA t a t t P t A       ] ) ~ [( 1 ] ~ [ ) ( ) ( ] ~ [ ) (

k



k

x x E x x E dx x dB x b x x P x B       ] ) ~ [( 1 ] ~ [ ) ( ) ( ] ~ [ ) (

k



7

CDF pdf mean kth moment

SLIDE 8

Laplace transform/moment generating fn

k k s k k k x s t s st

t ds s A d A e E s B t f E e E dt e t a s A ) 1 ( ) ( ) ( ] [ ) ( )] ( [ ] [ ) ( ) (

* ) ( * ~ * ~ *

      

    



8

SLIDE 9

Queueing system performance

Input variables: system defined by and Output variables: performance defined by

• 1. N(t)
• no. of customers in system at time t
• 2. wn

waiting time

• 3. sn

system time

HP: (statistical equilibrium,

stationarity)

t ~

x ~

N t N t    ) ( ,

9

SLIDE 10

N w s

k k k N k N

z P z Q z E N N E k N P P k N P k F



 

       ) ( ] [ ] [ ] [ ] [ ) (

) ( ] [ ] ~ [ ) ( ) ( ] ~ [ ) (

* ~

s W e E W w E dy y dW y w y w P y W

w s

    



) ( ] [ ] ~ [ ) ( ) ( ] ~ [ ) (

* ~

s S e E T s E dy y dS y s y s P y S

s s

    



10

SLIDE 11

Other performance variables:

I = idle period D = interdeparture time G = busy period Nq = no. of customers in queue

11

SLIDE 12

Kendall’s notation for queueing systems

A/B/m A/B/m/K/M no. of users

• no. of servers queue size

M exponential(Markovian) D deterministic

Er r-stage Erlangian G general Hr R-stage hyperexponential

12

SLIDE 13

• B. General results
• 1. Utilization factor

1 avg.) (on the use in capacity system

• f

fraction work do to system the

• f

capacity arrives

• rk

at which w rate avg.         C R

13

SLIDE 14

G/G/1

• Let us start with no assumptions on arrival and

service distribution and one single server

• It can be generally proven that:

sec / service

• f

rate avg. sec / arrivals no. avg.        x

1 where   x

14

SLIDE 15

G/G/m

• In case of multiple (m) servers:

) 1 where for except ( 1 requires Stability servers busy

• f

fraction avg. server each for time service avg. 1 ,                 D/D/m m m x

15

SLIDE 16

• 2. System time

W x T w E x E s E w x s       ] ~ [ ] ~ [ ] ~ [ ~ ~ ~

16

SLIDE 17

• 3. Little’s result

W N T N T N

q

        ,

The average number of customers in a queueing system is equal to the average arrival time of customers to that system, times the average time spent in that system

17

SLIDE 18

t) (0, interval the during system in the customers

• f

no. avg. t) (0, during arrived who customers all

• ver

averaged customer per time system ) area! grey the (i.e, t time to up system in the spent have customers all time total sec)

• (customer

) ( ) (    

t t t

N T ds s N t 

(t)

• (t)

N(t) t) (0, in departures no. ) ( t) (0, in arrivals no. ) (         t t

18

SLIDE 19

 

N ) ( ) ( ) ( N can But we t) (0, in system in customers no. Avg. ) ( N t) (0, in customer per time system Avg. ers sec/custom ) ( ) ( t) (0, in rate arrival Avg. sec ) ( ) (

t t t t t t

T t t t t t t t t T customers t t t                      

19

SLIDE 20

          m , : / / get also we and ) ( : t proven tha be can it Similarly, (q.e.d) then exist, lim and lim If                   m N N m G G W x T N N N x N W N T N T T

q q s s q t t

  t   t

20

SLIDE 21

• C. Poisson process

] ) ( in arrivals [ Pr , 1 , , ! ) ( ) ( 1 , , 1 ] ~ [ d, distribute lly exponentia t independen : times al Interarriv ,t k k e k t t P t t e t t P t

t k k t i

          

 



 

 

Pure birth system (see p. 60-63, 65,Vol. 1)

21

SLIDE 22

• 1. Derivation of the Poisson process

rate) (arrival intensity process t)

• (

t 1 ] in arrival 1 Pr[ 1 ] in arrival Pr[ t)

• (

t ] in arrival 1 Pr[                Δt Δt Δt

) ( ) ( ) ( ) ( ) ( ) ( ) ( ] 1 )[ ( ) (

1 1

t

• t

t P t t P t P t t P t

• t

t P t t P t t P

k k k k k k k

                   

 

   

22

SLIDE 23

1 ) ( that note ) ( ) ( ) 2 ( ) ( ] 1 )[ ( ) ( , 2 , 1 ) ( ) ( ) ( ) 1 (

1

              



P t P dt t dP t

• t

t P t t P k t P t P dt t dP

k k k

    

If I divide by «dt», I obtain: Similarly for P0: Solving the the first-order differential equation (2): then inserting P0(t) in (1) for k=1: then continuing by induction:

t

e t P

 

 ) (

, , ! ) ( ) (   



t k e k t t P

t k k 



t

te t P







 ) (

1

] ~ [ 1 ) ( ) !

• n!

distributi neg. exp. with (relation note Final * t t P e t P

t

   



23

SLIDE 24

• 2. Properties of the Poisson Process

2 2 1 ) 1 ( 1 ) 1 ( ) ( 2 ) (

1 1 ) s! (memoryles d distribute lly exponentia are times al Interarriv ) ( 1 ] ~ [ iv. ) , ( ] [ ) ( iii. ii. rate arrival avg. ) ( ) ( i.          

   

                

         

 

t e t a e t t P A(t) t e t dz t z dQ e z E z t P Q(z,t) t t t kP t N

t t z z t z z t k t N k k t N k k

24

SLIDE 25

Random process

• Poisson Process is a stochastic (or

random) process

• Now we will use some more advanced

stochastic processes (Markov chains)

• But… what is a stochastic process?

– It is «family» of random variables X(t) indexed by time t – A possible (intuitive) case is when the random process represents the sum of simple random variables at instant t

25

SLIDE 26

Example of a random process

 S1(t), S2(t) S3(t)…. are

random variables

 X(t) is random process  X(t,w) = Number of servers

busy at time t of realization w of the process= one realization of the process

 Assumptions

• Stationarity
• E[to,to+][X(t, w)] = A(t0, w)

= A (w)= A(w)

• Ergodicity
• A(w) = A

S1 S2 S3 S4 X(t) 4 3 2 1

´ ´ ´ ´ ´ ´ ´ ´

H

1

H

2

H

3

H

3

H

6

H

1

H

5

H

4

H

5

H

2

H

4

H

8

H

7

t H

7

H

3

H

4

H

7

H

6

H

5

H

6

H

7

H

8

H

8

t +  t

26

SLIDE 27

• D. Markov chains
• A random process is called a “chain” if the state space

is discrete (i.e., if X(t) can assume only discrete values) – Note: S = {0, 1, . . ., K} finite state S = {0, 1, . . .} infinite state (countable)

• A chain can be

– Discrete-time

• If t can assume only discrete values, i.e., if X(t) can change values on at

discrete instants in time

– Continuous-time

• If t can assume any continuous value
• A chain is a Markov chain if … see next slide

27

SLIDE 28

D.1. Discrete-time Markov chains

] [ ] , , , [

1 1 1 1 1 1    

      

n n n n n n

i X j X P i X i X i X j X P 

28

    N N S S Xn }, ..., , 1 , { ,

1) It’s a chain since: 3) It’s a Markov Chain since: 2) It’s a discrete-time chain since the time («x-axis») is slotted

SLIDE 29

State probabilities



   

i (n) i ) ( 1 ) ( ) ( ) (

1 ] [ ] [ define us Let      

n n n n n i

i X P

(One-step) transition probabilities



     

    j n ij n ij n n n n ij

n i P P P i X j X P P , 1 ] [ ] [ define us Let

) ( ) 1 ( ) 1 ( 1 ) 1 (

29

SLIDE 30

State equations

given , 2 , 1

) ( ) 1 ( ) 1 ( ) (

     

 

n P n

n n

Homogeneous chain



      

 j ij ij n n ij ij n ij

i P P P n i X j X P P n P P each for 1 ] [ ] [ time

• f

t independen i.e, ,

1 ) (

30

SLIDE 31

1 ) state" steady (" ies probabilit m equilibriu then exists, lim If 1 given , 2 , 1

) ( i ) ( i ) ( ) ( ) ( ) 1 ( ) (

       

 

 i i n n n n n n

P P n P            

  n

31

SLIDE 32

Proof of π(n) = π(n-1) P

32

SLIDE 33

Proof- Contd.

                            

       

... ... ] . . . . . . [ . . . . . .

2 1 ) 1 ( ) 1 ( 2 ) 1 ( 1 ) 1 ( ) 1 ( 2 ) 1 ( 2 1 ) 1 ( 1 ) 1 ( ) ( si i i i n s n n n si n s i n i n i n n i

P P P P P P P P         

i-th column of P 

33

SLIDE 34

Example

Discrete-time birth-death (arrival-departure) process with no waiting room. P(1 arrival at any time n) = a P(1 departure at any time n) = d s = {0,1} π = [π0 π1]

34

SLIDE 35

Example – Contd.

1 1 1 ) 1 ( ) (

) 1 ( ) 1 ( 1 1           d a d a P n P d d a a P

n n

                   



35

SLIDE 36

Example – Contd.

d a a d a d d a d a a d            

1 1 1 1

1 ) 1 ( 1         

36

SLIDE 37

D.2. Continuous-time Markov chains

    N N S S t X }, ..., , 1 , { , ) (

37

1) It’s a chain since: 3) It’s a Markov Chain since: 2) It’s a continuos-time chain since the times t can assume any real value

] ) ( ) ( [ ] ) ( ..., , ) ( ) ( [ ... .

1 1 1 1 1 2 1 n n n n n n n n

i t X j t X P i t X i t X j t X P t t t t any and n all For Def           

  

t1 t2 t3….. Finite or countable Real

SLIDE 38

Time spent in a state

r.v. geometric a is state a in spent Chain time Markov time

• discrete

a in : Note exp.) (negative 1 ] [ . . Chain Markov time

• continuos
• f

Property

t i i

i

e t P with v r a is S E state in time



 



    

38

SLIDE 39

• Neg. Exp. Is «Memoryless»

39

 

         

  

  

 

t T e e e e t T t T t t T t T t t T t t T t t T

t t t t t

                       

    

Pr 1 1 1 1 1 Pr Pr Pr Pr Pr Pr

   

Pr T  t t 0 T  t0

  Pr T  t

 

t

e t e

 t t0

 

SLIDE 40

D.2.a. General Case

D.2.a.1. Transition probabilities

i each for t q j i t t t t p t q t t t t p t q matrix rate transition t I t P t Q t t t p t P s t i s X j t X P t s p

j ij ij ij ii ii ij ij

) ( ) , ( lim ) ( 1 ) , ( lim ) ( ) ( lim ) ( )] , ( [ ) ( ] ) ( ) ( [ ) , (                      



If these limits do not exist, we do not have a continuous-time Markov chain

Δt→0 Δt→0 Δt→0

40

SLIDE 41

D.2.a.2 State Probabilities

 

        

t j j N j

ds s Q t t given t Q t dt t d N t t t t S j j t X P t

1

] ) ( exp[ ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( )], ( ... ) ( ) ( [ ) ( ] ) ( [ ) (           

41

SLIDE 42

D.2.b. Homogenous case

t Q ij ij ij ij ij

e t Q t dt t d q Q const q t q s

• f

indep t s s p t p ) ( ) ( ) ( ) ( ] [ . ) ( . ) , ( ) (            

42

SLIDE 43

Homogenous case – Contd.



    1 : ) ( ) ( lim ) (

j j j

Q

• f

t independen t exists it if state Steady     

t→∞

43

SLIDE 44

Models used in this course

Birth-death processes

• We want to apply the previously seen random processes

to model queues in telecom equipment

• The state changes only

– When a packet arrives (birth) – When a packet leaves (death)

• Basically you can only move between adjacent states

– State k  State k+1 (birth) – State k  State k-1 (death)

• Continuous time

– Packet arrive and depart at any time!

• Mathematical treatment follows in the next slides

44

SLIDE 45

• E. Continuous-time birth-death processes

aka arrival-departure processes (1) Continuous-time Markov chain dealing with a population

• f size N at time t

      L L S S k k t N P t P

k

} ..., , 1 , { ] ) ( [ ) (

(2) System state changes by at most one (up or down, or no change) in Δt

45

1 j

• k

if

, , 1 , 1

   

  j k k k k k k

p p dt p dt  

SLIDE 46

(3) Births and deaths independent

– Follows from Markovianity

(4) Transitions

rate death t

• t

k t N t t t in d exactly P t

• t

k t N t t t in d exactly P rate birth t

• t

k t N t t t in b exactly P t

• t

k t N t t t in b exactly P

k k k k k k

                                      ) ( 1 ] ) ( ) , ( [ ) ( ] ) ( ) , ( 1 [ ) ( 1 ] ) ( ) , ( [ ) ( ] ) ( ) , ( 1 [

Contd.

46

SLIDE 47

Transitions - State Eqns.

) ( 1 ) ( ) 3 ) ( ) ( ) ( ) ( ) ( ) 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1

10 1 00 , 1 1 , 1 1 ,

                  



    

L t P k t p t P t p t P t t P k t p t P t p t P t p t P t t P

L k k k k k k k k k k k k

47

• 3 (set of) equations regulate the birth-death process

SLIDE 48

State Eqns. – Contd.

)] ( )[ ( )] ( 1 )[ ( ) ( )] ( )[ ( )] ( )[ ( )] ( 1 )][ ( 1 )[ ( ) (

1 1 1 1 1 1

t

• t

t P t

• t

t P t t P t

• t

t P t

• t

t P t

• t

t

• t

t P t t P

k k k k k k k k

                             

   

     

48

• Solving of previous equations

*Similar derivation as seen in the Poisson process

SLIDE 49

State Eqns. – Contd.

) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) (

1 1 1 1 1 1

          

   

t k t P t P dt t dP k t P t P t P dt t dP

k k k k k k k k

     

49

*Solving these equations is quite complex, so, in practice, the approach used is the one shown in the next slide (inspection over a state diagram)

SLIDE 50

State-Transition Rate Diagrams

   

t P t P E

k k k k k 1 1 1 1

into Flow flow y probabilit

• f

Notion

   

   

   

t P E

k k k k

   

• f
• ut

Flow

         

... 1, 0, k E

• f
• ut

Flow

• E

into Flow

k 1 1 k 1 1

• k

k k

     

  

t P t P t P dt t dP

k k k k k

   

Note: a simple «inspection» technique to find the same equations

50

SLIDE 51

   

1,2,... k : P Hence, dP um) (equilibri balanced boundary across flows want we if

1 1 1 1 k k

    

    k k k k k k k k k

P P P P Note P t dt t    

51

SLIDE 52

(Generic M/M/1) Queue

... , 1 , ) ( , sec / sec /       k dt t dP t behavior Equilibrum jobs rate service jobs rate arrival

k k k

 

52

System Representation

SLIDE 53

  

             

                

1 1 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1

1 1 ... , , 1 1 , ) ( ) ( ; . 1

k k i i i k i i i k k k k k k k k k k k k k k

p p p p p p p p k p p k p p p p t P and dependent State                  

53

SLIDE 54

Classical M/M/1



 

                     

1

1 1 1 , ,

k k k k k k

p k p p k all        

54       k k+1   k-1       1 2

SLIDE 55

Classical M/M/1 – Contd.

) 1 ( ) 1 ( 1 1 1 1 1 , 1

1

                              

 

   

k p p p If

k k k k k k k k

                

55

SLIDE 56

2 2 2 2 2 1

) 1 ( ) ( 1 ) 1 ( 1 1 ) ( ) 1 (                                                             

      

               N k k N k k k k k k k k k k k k

p N k N d d d d d d k k k p k N

56

SLIDE 57

Contd.

      1 1 T T N

N W W W x T        1 1 / 1 / 1 1         

57

SLIDE 58

Contd.

1 ] [ ) 1 ( ] [ 1

2

          

 

   

       iff valid are results Above k N P p k N P N W N

k k i i k i i q q

58

SLIDE 59

M/M/m Queue Model

1      m

       m k m m k k

k

  

Equivalently,

59

SLIDE 60

Use state-dependent birth-death model to determine

,  k pk

3 2 3 1 3 2 1 2 1 2 1 1 1

3 ! 3 2 2                p p p p p p p p p

2 2 1 2 1 1 1 1

! ! ! p m m p p m m p p m p

m m m m m m    

     

 

   

      

1 1 1 1 1 1

1 1 ) ( ) ( 1 1 1 1

m k m k m k m k

ρ m! mρ k! mρ /m ρ ρ /m ρ m! ρ k! ρ p            m k p m m m k p k m p

k m k k

! ! ) (  

60

SLIDE 61

M/M/m Queue system characteristics

           m k p m m m k p k mp p

k m k k

! ! ) ( 



 

  

1

1 1 ) ( ) ( 1

m k m k

ρ m! mρ k! mρ p



 



m k k

p queueing p ] [ ) 1 ( ! ) ( ] [ p m m m p

m

                   m N p N p m N

s m q m

1 1     1    

s q

N x N W N T

Where

m

p

61

SLIDE 62

62

Backup slides

SLIDE 63

M/G/1 Queue Poisson arrival process:  customers/sec General service process:

k

x x x B , ), (

queueing discipline is FCFS

• 1. Pollaczek-Khinchin (P-K) relations

) 1 ( 2 ) 1 ( 2 1 ] [ ] [

2 2 2 2 2

                      x N W x N W x T x W x x E x x E x

63

SLIDE 64

Ex.

sec 33 . 6 33 . 4 2 53 . 2 ) 3 13 ( 4 . 8 . sec 33 . 4 3 13 ) 2 . ( 2 ) 3 13 )( 4 . ( ) 1 ( 2 8 . ) 2 )( 4 . ( sec 3 13 2 1 sec 2 sec / 4 .

2 2 3 1 2 2

                     



T W x T N W N W x W x dx x x x cust       

64

SLIDE 65

• 2. P-K transform relations

a.

z s sx k k k k

s B z B where z z B z z B z Q dx e x b x b L s B z Q Z p z p z Q

 

      

      

           

 

| ) ( ) ( ) ( ) 1 )( 1 ( ) ( ) ( ) ( )] ( [ ) ( )] ( [ , ) (

* * * * * 1

b.

 ) (y w

• prob. density fn. of waiting time

)], ( [ ) ( ) ( ) 1 ( ) ( )] ( [ ) (

* 1 * * *

      



y s W L y w s B s s s W y w L s W     ) (y s

• prob. density fn. of system time

)] ( [ ) ( ) ( ) ( ) 1 ( ) ( )] ( [ ) (

* 1 * * *

s S L y s s B s B s s s S y s L s S



        

65

SLIDE 66

y

e y s s s s s s s s s s s S

) ( 2 *

) ( ) ( ) 1 ( ) ( ) 1 ( ) /( ) 1 ( ) (

 

                     

 

                     

66

SLIDE 67

• 3. Modeling
• a. Imbedded Markov Chain



n

q

• no. customers left behind by

n

C 

n

v

• no. customers which enter during

n

x

 

         

  

,... 1 , , 1

1 1 1

n q q v q v q q

n n n n n n n

• Cont. –time Markov chain
• b. Tagged job

) 1 /( ) 1 ( ) ( ) (                 R W R W W R W x R W R x W R W

Residual work (residual life time)

67

SLIDE 68

 Residual Life - R x z Random Arrival Service process: b x

x x ( ), ,

2

F x P Z x

Z ( )

[ |   system busy at time of arrival] r = E[Z | system busy] Intuition: r = x 2 ? Wrong!

68

SLIDE 69

Chances are higher that x arrives during longer periods

f x density fn. for length of service period during which arrival occurs, given system busy f x x) Kx b(x) (x) f (x) d(x) K x b(x) dx = kx = 1, K = 1 x f (x) = x b(x) x r = x 2 x b(x) x d(x) = x 2x

Z Z Z Z 2

( ) ( ) (               

  

  

 

Residual Life - Cont’d

69

SLIDE 70

R = E[Z | system busy] E[system busy] = x x . x = 1 2 x W = 1 2 x x

2 2 2 2

2 1 2 1           ( ) ( )

 Markovian queueing Networks

• N-node interconnection of queueing systems
• queueing and service at each node
• Exponential service times at each node
• Model multi-service processes

Residual Life - Cont’d

70

SLIDE 71

1 3 4 2

r

11

r

12

r21 r31 r32 r22 r23 r44 r43 r34  1 1

43 44

  ( ) r r 1

31 32 34

   ( ) r r r  3

1. Open Networks N nodes Each node – single queue, servers Exponential service time

mi xi

i

 1  sec

71

SLIDE 72

• External arrivals – Poisson process (indep.) jobs/second
• = P[that a job which completes service at node will proceed next

to node ]

• P[that a job which completes service at node will leave

the network]

• Avg. arrival rate at node from both external sources ands other

nodes (including itself)

 i rij

i

j

1

1

 





r

ij j N

i

i 

i

rii r i

1

r i

2

rNi

 i

i

1 2 i N

  

i i j ji j N

r  





1

Flow Equations: = [

1 2 N

           ....... ] [ ....... ] [ ]

   

1 2 N ji

R r R

72

SLIDE 73

Jackson’s theorem (1957)

Let p(k k k p[k jobs at node 1, k jobs at node 2, ....., k jobs at node N] and p k p[k jobs at node i] Then p(k k k p k p k p k Each node in the network can be treated as an M / M / m queue, m 1

1 2 N 1 2 N i i i 1 2 N 1 1 2 2 N N i i

, ....., ) ( ) , ....., ) ( ) ( )........... ( )    

 

73

SLIDE 74

1 2 N-1 N

……………

 

N = Ni

i=1 N



  =

i i=1 N



T = N 

74

SLIDE 75

Example:

1 2 3

  r

12

r23

I/O I/O CPU

r

11

r22 r

11

02  . r

12

08  . r22 04  . r23 06  . 1 01

1

  . sec 1 005

2

  . sec 1 09

3

  . sec   1 job / sec

75

SLIDE 76

Node 1

         

1 11 1 1 1 1 1 1 1 1

1 02 125 0125 1 01429            r . . . . N1

Node 2

         

2 12 1 22 2 2 2 2 2 2 2 2

08 125 04 16667 00833 1 00909            r r . ( . ) . . . . N2

Node 2

        

3 22 2 3 3 3 3 3

06 16667 1 09 1 09 9 2338 9 23            r . ( . ) . . . . sec N N = N + N + N T = N

3 1 2 3

76

SLIDE 77

1 4 3 2

r

12

r21 r

14

r42 r22 r23 r33 r43

• 2. Closed Networks

N nodes; K jobs circulating through the network No external arrivals or departures Each node – single queue, servers Exponential service time

mi xi

i

 1  sec r24 r32

      

i ij N

r     

 

1

1 2

for each i K K R = [r (flow vector) R

j=1 N i i=1 N ij]

[ .......... ]

77

SLIDE 78

Gorjon and Newell (1967)

p(k k k 1 G(K) x k {x satisfy x x i = 1,2,....., N G(K) = x k with k = (k ,k ,.......,k ) and A = set of all k vectors for which k + k +.....+k = k (3) k k

1 2 N i k i i i i i j i k i i k 1 2 N 1 2 N i i

i i

, ....., ) ( ) ( ) } ( ) ( ) ( )   

   

 

    

i N j ji j N i N A

where r

1 1 1

1 2

 

i i i i i k m i i i i i

, k m m m , k m Utilization at node i: x m i = 1,2, ..... , N

i i

! !      





78

SLIDE 79

Example: (Application-interactive computing) Multi-Processor

. . . . . . . . . . . . . .

Node 1 Nodes 2,3, …….N K terminals k jobs

      N = avg. "think" time at each terminal (exp. distribution think times) T = K seconds T = avg. responce time

1

 

79

SLIDE 80

) , ( M/M/1 2.

k k

     

1 k 1 k

P 1,2,... k P m Equilibriu

 

  

k k

P P    



 

                    

k 2 1 2 1

1 P subject to P P to leads This

k k k

P P P P P         

80