Cheng-Fu Chou, CMLab, CSIE, NTUCheng-Fu Chou, CMLab, CSIE, NTU

M/G/1

Cheng-Fu Chou

P. 2

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Residual Life

Inter-arrival time of bus is exponential w/ rate while hippie arrives at an arbitrary instant in time

Question: How long must the hippie wait, on the average , till the bus comes along?

Answer 1: Because the average inter-arrival time is 1/, therefore 1/2

P. 3

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Residual Life (cont.)

Answer 2: because of memoryless, it has to wait 1/

General Result

1

21

1

1

1

**

1

2 ly,particular

)1(

)(1)(

)(1)(ˆ

)()(

m

mr

mn

mr

sm

sFsF

m

yFyf

dxxkxfdxxf

nn

X

X

P. 4

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Derivation

P. 5

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 6

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

M/G/1

M/G/1– a(t) = e –t

– b(t) = general

Describe the state [N(t), X0(t)]– N(t): the no. of customers present at time t– X0(t): service time already received by the

customer in service at time t

Rather than using this approach, we use “the method of the imbedded Markov Chain”

P. 7

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Imbedded Markov Chain [N(t), X0(t)] Select the “departure” points, we therefore

eliminate X0(t) Now N(t) is the no. of customer left behind by a

departure customer. (HW)– For Poisson arrival pk(t) = rk(t)– If in any system (even in non-Markovian) where

N(t) makes discontinuous changes in size (plus or minus) one, thenork = dk = prob[departure leaves k customers

behind] – Therefore, for M/G/1

ork = dk = pk

P. 8

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 9

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 10

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Mean Queue Length

P. 11

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 12

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 13

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 14

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 15

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 16

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 17

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

This is the famous Pollaczek – Khinchin Mean Value Formula

P. 18

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Examples

P. 19

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Mean Residual Service Time

Wi: waiting time in queue of the i-th customer Ri: residual service time seen by the i-th customer Xi: service time of the i-th customer Ni: # of customers found waiting in the queue by the i-th

customer upon arrival–

}{ lim R as defined time,residualmean R where

,1

R W

obtain we, i aslimit theTaking

}{}{

)}|({}{}{

i

1

1

i

Q

i

ij

i

Nijii

i

Nijjii

RE

N

NExRE

NXEEREWE

XRW

i

i

P. 20

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

)r(argument graphical aby R calculatecan we

R?get tohow :Question

where,1

1

N 1

have we

result, sLittle'by ),( existslimit that Assume

Q

RW

WR

λWμ

R

RW

WNQ

P. 21

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Residual Service Time

x1

x1

x2 xM(t)

r

time t

P. 22

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

)1(2

)1(2

2

1

)(lim

)(lim

2

1)(

1lim

exist limits theassuming

)(

)(

2

1

2

11)(

1 so,

customerth -i of timeservice theis x

t][0, within completion service of # is M(t) where

2

11)(

1

22

2

2

)(

1

2

0

)(

1

2

2)(

10

i

2)(

10

xN

xxwxT

x

tM

x

t

tMdr

tR

tM

x

t

tMx

tdr

t

xt

drt

tM

ii

tt

t

t

tM

ii

i

tM

i

t

i

tM

i

t

P. 23

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Distribution of Number in the System

P. 24

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

P. 25

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Ex

Q(z) for M/M/1

P. 26

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Sol.

P. 27

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Waiting Time Distribution

P. 28

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Ex

Response time distribution for M/M/1 Waiting time distribution for M/M/1

P. 29

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Response Time Distribution

P. 30

Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU

Waiting Time Distribution