Queueing theory primer - UC Davis: Networks...
Transcript of Queueing theory primer - UC Davis: Networks...
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Queueing theory
primer
Lecturer: Massimo Tornatore Original material prepared by:
Professor James S. Meditch
Typesetter: Dr. Anpeng Huang
Courtesy of: Prof. Biswanath Mukherjee 1
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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
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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
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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?)
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A. Notation and terminology
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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~
1n
s~
x~
w~
*Note that these distributions do not depend by n (same distribution for all arrivals/services) 6
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Arrival process Service process
kttE
ttE
dt
tdAta
ttPtA
])~
[(
1]
~[
)()(
]~
[)(
k
kxxE
xxE
dx
xdBxb
xxPxB
])~[(
1]~[
)()(
]~[)(
k
7
CDF
mean
kth moment
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Laplace transform/moment generating fn
kk
sk
kk
xs
tsst
tds
sAdA
eEsB
tfEeEdtetasA
)1()(
)0(
][)(
)]([][)()(
0
*)(*
~*
~
0
*
8
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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~
NtNt )(,
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N w s
k
k
k
N
k
N
zPzQzE
NNE
kNPP
kNPkF
0
)(][
][
][
][)(
)(][
]~[
)()(
]~[)(
*~
sWeE
WwE
dy
ydWyw
ywPyW
ws
)(][
]~[
)()(
]~[)(
*~
sSeE
TsE
dy
ydSys
ysPyS
ss
10
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Other performance variables:
I = idle period
D = interdeparture time
G = busy period
Nq = no. of customers in queue
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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
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B. General results
1. Utilization factor
10
avg.) (on the usein capacity system offraction
workdo tosystem theofcapacity
arrivesork at which w rate avg.
C
R
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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 of rate avg.
sec / arrivals no. avg.
x
1
where
x
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G/G/m
• In case of multiple (m) servers:
)10 wherefor except (
10 requiresStability
serversbusy offraction avg.
servereach for timeservice avg. 1
,
D/D/m
mm
x
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2. System time
WxT
wExEsE
wxs
]~[]~[]~[
~~~
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3. Little’s result
WNTN
TN
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
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t)(0, interval theduring system in the customers of no. avg.
t)(0, during arrived who
customers allover averagedcustomer per timesystem
)area!grey the(i.e, t time toup system in thespent have
customers all time totalsec)-(customer )()(0
t
t
t
N
T
dssNt
0(t)- (t)N(t)
t)(0,in departures no.)(
t)(0,in arrivals no.)(
t
t
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N
)(
)()(Ncan But we
t)(0,in systemin customers no. Avg. )(
N
t)(0,in customer per timesystem Avg. erssec/custom )(
)(
t)(0,in rate arrival Avg. sec
)()(
t
t
t
tt
t
T
t
t
t
t
t
t
t
tT
customers
t
tt
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m , ://
get also weand
)(
:tproven tha becan it Similarly,
(q.e.d) then exist, lim and lim If
mNNmGG
WxT
NNNxN
WN
TNTT
q
qs
s
q
ttt t
20
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C. Poisson process
])0(in arrivals[ Pr
,1,0,!
)()(
1,0,1]
~[
d,distributelly exponentia
tindependen : timesalInterarriv
,tk
kek
ttP
ttettP
t
tk
k
t
i
Pure birth system (see p. 60-63, 65,Vol. 1) 21
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1. Derivation of the Poisson process
rate) (arrivalintensity process
t)o(t1
]in arrival 1Pr[1]in arrival 0Pr[
t)o(t]in arrival 1Pr[
ΔtΔt
Δt
)()()()()(
)()(]1)[()(
1
1
tottPttPtPttP
tottPttPttP
kkkk
kkk
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1)0( that note )()(
)2(
)(]1)[()(
,2,1 )()()(
)1(
000
00
1
PtPdt
tdP
tottPttP
ktPtPdt
tdPkk
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:
tetP )(0
0,0,!
)()( tke
k
ttP t
k
k
ttetP )(1
]~
[1)( )!on!distributi neg. exp.with (relation note Final* 0 ttPetP t
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2. Properties of the Poisson Process
2
2
1
)1(
1
)1(
0
)(
2
)(
0
1
1
)s!(memoryles ddistributelly exponentia are timesalInterarriv
)(
1]~
[ iv.
),(
][)( iii.
ii.
rate arrival avg.
)()( i.
t
eta
ettPA(t)
tetdz
tzdQ
ezEztPQ(z,t)
t
ttkPtN
t
t
z
zt
z
zt
k
tNk
k
tN
k
k
24
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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
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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
0
´
´
´
´
´
´
´
´
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 0
H 7
H 3 H 4 H 7 H 6 H 5
H 6 H 7 H 8
H 8
t 0 + t
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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
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D.1. Discrete-time Markov chains
][
],,,[
11
001111
nnn
nnn
iXjXP
iXiXiXjXP
28
NNSSXn },...,,1,0{,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
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State probabilities
i
(n)
i
)(
1
)(
0
)()(
1
][ ][ define usLet
nnn
n
n
i iXP
(One-step) transition probabilities
j
n
ij
n
ij
n
nn
n
ij
niPPP
iXjXPP
, 1 ][
][ define usLet
)()1()1(
1
)1(
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State equations
given ,2,1 )0(
)1()1()(
n
P nnn
Homogeneous chain
j
ijij
nnij
ij
n
ij
iPPP
niXjXPP
nPP
each for 1 ][
][
timeoft independen i.e, ,
1
)(
30
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1
)state"steady (" iesprobabilit mequilibriu
thenexists, lim If
1 given
,2,1
)(
i
)(
i
)0(
)0()(
)1()(
i
i
n
n
nn
nn
P
P
nP
n
31
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Proof of π(n) = π(n-1) P
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Proof- Contd.
...
...]......[
......
2
1
0
)1()1(
2
)1(
1
)1(
0
)1(
2
)1(
21
)1(
10
)1(
0
)(
si
i
i
i
n
s
nnn
si
n
s
i
n
i
n
i
nn
i
P
P
P
P
P
PPP
i-th column of P
33
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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
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Example – Contd.
101
100
)1()(
)1(
)1(
1
1
da
da
Pn
P
dd
aaP
nn
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Example – Contd.
da
a
da
d
d
a
d
aad
10
010
0101
1)1(1
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D.2. Continuous-time Markov chains
NNSStX },...,,1,0{,)(
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
])()([])(...,,)()([
...0.
1111
121
nnnnnn
nn
itXjtXPitXitXjtXP
ttttanyandnallForDef
t1 t2 t3…..
Finite or
countable
Real
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Time spent in a state
r.v. geometric a is state ain
spent Chain time Markov time-discrete ain :Note
exp.) (negative 1][
..
Chain Markov time-continuos ofProperty
t
i
i
ietP
withvraisSEstateintime
38
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Neg. Exp. Is «Memoryless»
39
tT
ee
ee
tT
tTttT
tT
ttTttTttT
t
t
ttt
Pr
111
11
Pr
PrPr
Pr
Pr Pr
0
00
0
00
0
0000
Pr T t t 0 T t0 Pr T t
t 0
et
e t t0
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D.2.a. General Case
D.2.a.1. Transition probabilities
ieachfortq
jit
tttptq
t
tttptq
matrixratetransitiont
ItPtQ
tttptP
stisXjtXPtsp
j
ij
ij
ij
iiii
ij
ij
0)(
),(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
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D.2.a.2 State Probabilities
t
j
j
N
j
dssQt
tgiven
tQtdt
td
Ntttt
SjjtXPt
0
10
])(exp[)0()(
1)()0(
)()()(
)],(...)()([)(
])([)(
41
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D.2.b. Homogenous case
tQ
ij
ijij
ijij
etQtdt
td
constqtq
sofindeptssptp
)0()()()(
][
.)(
.),()(
42
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Homogenous case – Contd.
1
0
:)0(
)(lim
)(
j
jj
Q
oftindependen
t
existsitifstateSteady
t→∞
43
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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
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E. Continuous-time birth-death processes aka arrival-departure processes
(1) Continuous-time Markov chain dealing with a population
of size N at time t
LLS
SkktNPtPk
}...,,1,0{
])([)(
(2) System state changes by at most one (up or down, or no
change) in Δt
45
1j-k if 0,
,1
,1
jk
kk
kkk
p
pdt
pdt
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(3) Births and deaths independent
– Follows from Markovianity
(4) Transitions
ratedeath
totktNtttindexactlyP
totktNtttindexactlyP
ratebirth
totktNtttinbexactlyP
totktNtttinbexactlyP
k
k
k
k
k
k
)(1])(),(0[
)(])(),(1[
)(1])(),(0[
)(])(),(1[
Contd.
46
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Transitions - State Eqns.
)(1)( )3
0 )()()()()( )2
1 )()(
)()()()()( )1
0
1010000
,11
,11,
LtP
ktptPtptPttP
ktptP
tptPtptPttP
L
k
k
kkk
kkkkkkk
47
- 3 (set of) equations regulate the birth-death process
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State Eqns. – Contd.
)]()[()](1)[()(
)]()[()]()[(
)](1)][(1)[()(
11000
1111
tottPtottPttP
tottPtottP
tottottPttP
kkkk
kkkk
48
- Solving of previous equations
*Similar derivation as seen in the Poisson process
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State Eqns. – Contd.
0
0)()()(
1
)()()()()(
11000
1111
t
ktPtPdt
tdP
k
tPtPtPdt
tdPkkkkkkk
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)
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State-Transition Rate Diagrams
tPtPE kkkkk 1111 into Flow
flowy probabilit ofNotion
tPE kkkk ofout Flow
... 1, 0,k
E ofout Flow - E into Flow
k11k11-k
kk
tPtPtP
dt
tdP
kkkk
k
Note: a simple «inspection» technique to find the same equations
50
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1,2,...k
:
P Hence, 0dP
um)(equilibri balancedboundary across flows want weif
11
11
kk
k
k
kk
kkkk
k
PP
PP
Note
Ptdt
t
51
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(Generic M/M/1) Queue
...,1,00)(
,
sec/
sec/
kdt
tdP
tbehaviorEquilibrum
jobsrateservice
jobsratearrival
k
k
k
52
System Representation
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1
1
0 1
0
1
0 1
0
0
21
1020
1
01
0
1100
1111
1
1
...,,
1
00
1,)(0
)(;.1
k
k
i i
i
k
i i
ik
k
k
kkkkkkk
kkkk
p
pp
pppp
p
kpp
kppp
ptPanddependentState
53
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Classical M/M/1
1
0
0
1
1
1
,,
k
k
k
k
kk
p
kpp
kall
54
k k+1
k-1
10 2
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Classical M/M/1 – Contd.
0)1(
)1(1
10
1
11 ,1
0
01
kp
pp
If
k
k
k
k
k
k
k
k
55
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2
2
0
22
20
00 0
1
0 0
)1(
)(
1
)1(1
1
)(
)1(
N
k
kN
k
k
k
k
k k
kk
k k
k
k
pNk
N
d
d
d
d
d
dkk
kpkN
56
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Contd.
1
1
TTN
NW
WWxT
1
1
/
1
/11
57
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Contd.
10
][
)1(][
1
2
iffvalidareresultsAbove
kNP
pkNP
NWN
k
ki
i
ki
i
58
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M/M/m Queue Model
1
m
mkm
mkkk
0
Equivalently,
59
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Use state-dependent birth-death model to determine 0, kpk
320
3
13
210
2
12
10011
3!3
22
ppp
ppp
ppp
02
2
12
0
1
11
01
!
!
!
pmm
p
pmm
p
pm
p
m
m
m
m
m
m
1
0
11
0 1
11
0
1
1)()(
1
1
1
1
m
k
mk
m
k
mk
ρm!
mρ
k!
mρ
/mρρ
/mρm!
ρ
k!
ρp
mkpm
m
mkpk
m
pkm
k
k
0
0
!
0!
)(
60
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M/M/m Queue system characteristics
mkpm
m
mkpk
mp
pkm
k
k
0
0
!
0!
)(
1
0
0
1
1)()(
1m
k
mk
ρm!
mρ
k!
mρp
mk
kpqueueingp ][
0)1(!
)(][ p
m
mmp
m
mNpN
pmN
smq
m
1
1
1 sq N
xN
WN
T
Where
mp
61
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62
Backup slides
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M/G/1 Queue
Poisson arrival process: customers/sec
General service process: kxxxB ,),(
queueing discipline is FCFS
1. Pollaczek-Khinchin (P-K) relations
)1(2
)1(2
1][][
22
2
22
xNWxN
WxTx
W
xxExxEx
63
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Ex.
sec33.633.42
53.2
)3
13(4.08.0
sec33.4
3
13
)2.0(2
)3
13)(4.0(
)1(2
8.0)2)(4.0(
sec3
13
2
1
sec2
sec/4.0
2
2
3
1
22
TWxT
N
WN
W
xW
x
dxxx
x
cust
64
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2. P-K transform relations
a.
zs
sx
k
k
k
k
sBzB
where
zzB
zzBzQ
dxexbxbLsB
zQZpzpzQ
|)()(
)(
)1)(1()()(
)()]([)(
)]([,)(
**
*
*
0
*
1
0
b.
)(yw prob. density fn. of waiting time
0)],([)(
)(
)1()(
)]([)(
*1
*
*
*
ysWLyw
sBs
ssW
ywLsW
)(ys prob. density fn. of system time
)]([)(
)()(
)1()(
)]([)(
*1
*
*
*
sSLys
sBsBs
ssS
ysLsS
65
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yeys
s
s
ss
s
sss
ssS
)(
2
*
)()(
)1(
)(
)1(
)/(
)1()(
66
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3. Modeling
a. Imbedded Markov Chain
nq no. customers left behind by nC
nv no. customers which enter during nx
,...1,0,
0
01
1
1
1
nq
qv
qvqq
n
nn
nnn
n
Cont. –time Markov chain
b. Tagged job
)1/(
)1(
)(
)(
RW
RW
WRWxRW
R
xWRW
Residual work (residual life time)
67
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Residual Life - R
x
z
Random Arrival
Service process:
b x x x( ), , 2
F x P Z xZ ( ) [ | system busy at time of arrival]
r = E[Z | system busy]
Intuition: r =x
2 ? Wrong!
68
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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
Z00
Z
0
2
( )
( ) (
Residual Life - Cont’d
69
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R = E[Z | system busy] E[system busy]
=x
x . x =
1
2 x
W =
1
2x
x
22
22
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
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1
3
4
2
r11
r12
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
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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 ij
11
rijj
N
ii
i
rii
r i1
r i2
rNi
i
i
1
2i
N i i j jij
Nr
1
Flow Equations:
= [ 1 2 N
....... ]
[ ....... ]
[ ]
1 2 N
jiR r
R
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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
, ....., )
( )
, ....., ) ( ) ( )........... ( )
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1
2
N-1
N
……………
N = N ii=1
N
= ii=1
N
T =N
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Example:
1 2 3
r12 r23
I/O I/O CPU
r11 r22
r11 02 . r12 08 .
r22 04 . r23 06 .
101
1 . sec
10 05
2 . sec
10 9
3 . sec
1 job / sec
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Node 1
1 11 1 1 1
11
1
1
1
1 0 2 125
01251
01429
r . .
. . N1
Node 2
2 12 1 22 2 2 2
22
2
2
2
08 125 0 4 16667
0 08331
0 0909
r r . ( . ) . .
. . N2
Node 2
3 22 2
33
3
3
3
0 6 16667 1
0 91
0 9
9 2338
9 23
r . ( . )
. .
.
. sec
N
N = N + N + N
T =N
3
1 2 3
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1
4
3
2 r12
r21
r14
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 mixi
i
1
sec
r24
r32
i ij
N
r
0 1
1 2
for each i
K K R = [r
(flow vector)
R
j=1
N
ii=1
N
ij ]
[ .......... ]
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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) kk
1 2 Ni
k
i i
i
i i j
i
k
i ik
1 2 N
1 2 N
i i
i
i
, ....., )( )
( ) }
( )( )
( )
i
N
j jij
N
i
N
A
where
r
1
1
1
1
2
i i i
i i
k m
i i
ii
i
, k m
m m , k m
Utilization at node i: x
m i = 1,2, ..... , N
i i
!
!
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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
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),( M/M/1 2. kk
1k
1k
P
1,2,...k
P
mEquilibriu
k
k
P
P
0
k
0
0
2
12
01
1P
subject to
P
P
toleads This
k
k
k PP
PP
P
80