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. 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. 17
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
This is the famous Pollaczek – Khinchin Mean Value Formula
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. 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