1 machine interference problem: introduction n machines each may break down and join the repair’s...
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Machine interference problem: introduction
N machines Each may break down and join the repair’s man queue
Operation time Exponentially distributed with rate λ
Repair time Exponentially distributed with rate μ
Nmachines
Repair’s man queue
1/μ1/λ
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Machine interference problem: Introduction (cont’d)
Each of the N machines can be thought of As being a server
You get a 2 node closed queuing network As long as the machine holds a client called token
The machine is operational
# tokens = # machines
4 customers (tokens)
1/λ
1/μ
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Machine interference problem: history Early computer systems
Multiple terminals sharing a computer (CPU) Jobs are shifted to the computer
Jobs run according to a Time Sharing idea
Main performance issue How many terminals can I support so that
Response time is in the order of ms
=> machine interference problem Operational => either thinking or typing
Hitting the return key => machine breaks down
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Machine interference problem: assumptions
Problem (assumptions) Operative
Mean = 1/λ
Repair time Mean = 1/μ
Repair queue FIFO
Finite population of customers
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Machine interference problem: solution
Birth and death equations
What about P0?
00
001
110
)!(
!))1()...(1(.
......
))1(...()1.(
...
...
,...,1,0;)(
;0
PnN
NnNNNPP
PnNNN
PP
NnnN
Nn
nn
n
n
nn
n
n
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Normalizing constant
N
n
n
N
nNN
P
PPPP
0
0
210
)!(!
1
1...
Rate diagram#1 State: # of broken down machines
Rate diagram#2 (including more redundancy) State: # of both active and broken down machines
0 1
μ
Nλ (N-1)λ
….
N,0 N-1,1
μ
Nλ (N-1)λ
….
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Machine interference problem: performance measures
Mean repair’s man queue length
Mean # customers in the entire system
Mean waiting time (Little’s theorem) What is the arrival rate to the repair’s man queue?
)1().1( 01
PNPnLN
nnq
)1( 0PLL q
W
qq WL
WL
.
.
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Arrival rate to repair’s man queue and waiting time Arrival rate to repair’s man queue
Mean waiting time in repair’s man queue
Mean waiting in the entire repair’s man system
).(..
........
)..(
0000
00
LNLN
PnPNPnPN
PnNP
N
nn
N
nn
N
nn
N
nn
N
nn
N
nnn
qqq LLN
LW .)(
11
LLN
LW .).(
11
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Single machine: analysis
Cycle thru which goes a machine
Mean cycle time
Rate at which a machine completes a cycle
Rate at which all machines complete their cycle
Operational
Wait
Repair
W1
W11
W
N
1
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Production rate
# of repairs per unit time Production rate
= rate at which you see machines Going in front of you
)1.( 0P
1
)1.(
)1.()1.(
)1.()1.()1.(1
0
00
000
P
NW
PNPW
PWPNPW
N
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Mean repair’s man queue length Lq
)1.()(
)1.()()1).((
)1(
)1(
)]1.().[.()1.(..
1
)1.()..(
)..(.
0
00
0
0
00
0
PNL
PNPN
PLL
PNL
PNLNPL
P
NLNL
WLNLWL
q
q
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Normalized mean waiting time
W (mean waiting time) is given by
r = average operation time/average repair time
Normalized mean waiting time
W = 30 min, 1/μ=10 min => normalized WT = 3 repair times
)1(.
1
)1.( 00 P
NW
P
NW
timerepairaverage
timeoperationaverager
__
__
/1
/1
timeservicemean
timewaitingmeanWW
__
__
/1
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Normalized mean waiting time: analysis
Plot the normalized waiting time As a function of N (# machines)
N=1 => W=1/μ => P0 = r/(1+r)
N is very large =>
Normalized mean waiting time Rises almost linearly with the # of machines
rP
N
P
NW
)1()1(.
00
rNWP .00
μW
N
1
N-r
1+r
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Mean number of machines in the system L
Plot L as a function of N
N=1 => P0 = r/(1+r) => L = 1/(1+r)
N is very large L = N - r
)1.( 0PrNL
L
N
1/(1+r)
N-r
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Examples
Find the z-transform for Binomial, Geometric, and Poisson distributions
And then calculate The expected values, second moments, and variances
For these distributions
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Z-transform: application in queuing systems
X is a discrete r.v. P(X=i) = Pi, i=0, 1, …
P0 , P1 , P2 ,…
Properties of the z-transform g(1) = 1, P0 = g(0); P1 = g’(0); P2 = ½ . g’’(0)
, +
0
)(i
iizPzg
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Binomial distribution
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Geometric distribution
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Poisson distribution
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Problem I Consider a birth and death system, where:
Find Pn
nn
n
nk nPP
n
nPP
)1(...2.1
)1...(3.2
...
...000
21
10
n
n nP
112
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Problem I (cont’d)
Find the average number of customers in system
03
22
0 1
2
1
21)1(1..
n
n
nn nnPnN
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Problem II In a networking conference
Each speaker has 15 min to give his talk Otherwise, he is rudely removed from podium
Given that time to give a presentation is exponential With mean 10 min
What is the probability a speaker will not finish his talk? E[X] = 1/λ = 10 minutes => λ = 1/10 Let T be the time required to give a presentation: a
speaker will not manage to finish his presentation if T exceeds 15 minutes.
P(T>15) = e-1.5
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Problem III Jobs arriving to a computer
require a CPU time exponentially distributed with mean 140 msec.
The CPU scheduling algorithm is quantum-oriented job not completing within 100 msec will go to back of queue
What is the probability that an arriving job will be forced to wait for a second quantum?
Of the 800 jobs coming per day, how many Finish within the first quantum>
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Problem IV A taxi driver provides service in two zones of a city.
Customers picked up in zone A will have destinations in zone A with probability 0.6 or in zone B with probability 0.4.
Customers picked up in zone B will have destinations in zone A with probability 0.3 or in zone B with probability 0.7.
The driver’s expected profit for a trip entirely in zone A is 6$; for a trip in zone B is 8$; and for a trip involving both zones is 12$.
Find the taxi driver’s average profit per trip. Hint: condition on whether the trip is entirely in zone A, zone B, or in
both zones.
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Problem V
Suppose a repairman has been assigned The responsibility of maintaining 3 machines.
For each machine The probability distribution of running time
Is exponential with a mean of 9 hours The repair time is also exponential
With a mean of 12 hrs
Calculate the pdf and expected # of machines not running
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Problem V (continued)
As a crude approximation It could be assumed that the calling population is infinite
=> input process is Poisson with mean arrival rate of 3 / 9 hrs
Compare the results of part 1 to those obtained from M/M/1 model and an M/M/1/3 model
Which one is a better approximation?