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Transcript of Network Design and Analysis-----Wang Wenjie Queueing System IV: 1 © Graduate University, Chinese...
Network Design and Analysis-----Wang Wenjie Queueing System IV: 1
© G
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Network Design and Analysis
Wang Wenjie
Network Design and Analysis-----Wang Wenjie Queueing System IV: 2
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Queueing System IV
Discrete-Time Queuing Systems
Network Design and Analysis-----Wang Wenjie Queueing System IV: 3
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1. Discrete-Time Queuing Systems
1. 1. Motivation
1. 2. The Bernoulli Process
1. 3. Geo/Geo/1 Systems
1. 4. Geo/Geo/1/N Systems
1. 5. Simple ATM Queuing Systems
Network Design and Analysis-----Wang Wenjie Queueing System IV: 4
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1.1 Motivation
Some queuing systems operate on a slotted
time basis: DT Systems
Network Design and Analysis-----Wang Wenjie Queueing System IV: 5
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Model
• All events (arrivals, departures) must occur at integer multiples of the slot time TS
• Implication: all service times must be multiples of TS
• For convenience, often use normalized time
• Example: ATM
– Fixed-size cells imply all service times equal 1 time
slot (on a given link)
Network Design and Analysis-----Wang Wenjie Queueing System IV: 6
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Discrete-Time Markov Chain
• Let j(n) = P[ X(n) = j ], j= 0, 1, …
• This denotes the probability of being in state j at time n , so
1j
j
On Off1-
1-
Network Design and Analysis-----Wang Wenjie Queueing System IV: 7
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Transition Probability Matrix
Network Design and Analysis-----Wang Wenjie Queueing System IV: 8
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Properties of P
• A square matrix whose dimension is the size of the state space
• Every row adds up to 1
(n+m) = (n) Pm
• Therefore, if we know the initial state pmf (0), we can get the state pmf at any time m.
• If DTMC is stationary, in steady-state:
P
Network Design and Analysis-----Wang Wenjie Queueing System IV: 9
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Exercise 1.1
An urn initially contains 5 black balls and 5 white balls. The following experiment is repeated indefinitely:
• A ball is drawn from the urn
• If the ball is white it is put back in the urn
• If the ball is black it is left out
X(n) is the number of
black balls after n
draws from the urn
Network Design and Analysis-----Wang Wenjie Queueing System IV: 10
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Exercise 1.1(Cont’d)
1. Draw the Markov chain and find the transition probabilities.
2. Find the matrix P.
3. Find the probability that there are 4 black balls in the urn after 2 draws.
Network Design and Analysis-----Wang Wenjie Queueing System IV: 11
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1.2. The Bernoulli Process
• Discrete-time analogous to Poisson process
– P[1 arrival during a time slot] = a
– P[0 arrivals during a time slot] = 1-a
• Mean # of arrivals during a time slot = ____
• Motivation: synchronous high-speed packet switches where at most one packet can be transmitted over a link during a slot
Similarities to the Poisson process•Memoryless•Multiplexing and demultiplexing of a Bernoulli process still result in Bernoulli processes.
Network Design and Analysis-----Wang Wenjie Queueing System IV: 12
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Geometric Distribution (1/2)
• P[next arrival occurs within k time slots]
= P[k-1 empty slots] P[arrival] = (1-p)k-1 p
Geometric distribution
• Poisson process Exponential Interarrival Time
• Bernoulli process Geometric Interarrival Time
Network Design and Analysis-----Wang Wenjie Queueing System IV: 13
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Geometric Distribution (2/2)
• Geometric distribution has the memoryless property
• Mean : 1/p
• Variance : (1-p)/p2
Network Design and Analysis-----Wang Wenjie Queueing System IV: 14
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Binomial Distribution
• N slots with i arrivals in some sequence :
P(sequence)=pi(1-p)N-i
• There are i arrivals in N slots for some value of p, the binomial distribution is :
b(i ,N, p)= CiNpi(1-p)N-i
• Mean : Np
Network Design and Analysis-----Wang Wenjie Queueing System IV: 15
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1.3. Geo/Geo/1 Systems
• Bernoulli arrivals, geometric service times
(DT analog to M/M/1)
Network Design and Analysis-----Wang Wenjie Queueing System IV: 16
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Model
• Let a = P[arrival]
• Let s = P[service completion] (departure)
Note that this is a discrete time Markov chain, so these are transition probabilities .NOT transition rates.
Network Design and Analysis-----Wang Wenjie Queueing System IV: 17
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Local Balance Equations
Analogous to in M/M/1 system
0110 )1(
)1()1()1( p
as
sapaspsap
0
2
221 )1(
)1()1()1( p
as
sapaspsap
r
Network Design and Analysis-----Wang Wenjie Queueing System IV: 18
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State Probabilities
0prp nn
rpr
prpp
n
n
nn
111 0
0
000
nn rrp )1(
Network Design and Analysis-----Wang Wenjie Queueing System IV: 19
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Average # of Customers
If 0< r <1 then:
00
)1(][n
n
n
n nrrnpnE
r
rnE
1][
Network Design and Analysis-----Wang Wenjie Queueing System IV: 20
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Traffic Intensity
s
aar
apserveridlep
1)1)(1(
)1(][ 0
s
a
Network Design and Analysis-----Wang Wenjie Queueing System IV: 21
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Mean Delay
)1(
][][
ra
r
a
nEE
• System throughput (customers/slot time) = a =
• Apply Little’s law
Network Design and Analysis-----Wang Wenjie Queueing System IV: 22
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1.4. Geo/Geo/1/N Systems
• Similar to Geo/Geo/1, except for state N
Network Design and Analysis-----Wang Wenjie Queueing System IV: 23
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Local Balance Equations
• As before, local balance yields
• However Now:
Nnprp nn for 0
01
1
)1()1(
)1(
parps
sap
spsap
NNN
NN
Network Design and Analysis-----Wang Wenjie Queueing System IV: 24
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State Probabilities
1
0
0)1(
1N
n
Nn arrp
)1(1
110 rarr
rp
NN
Network Design and Analysis-----Wang Wenjie Queueing System IV: 25
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Delay
• P[block] = P[N in system] = N
• So, throughput is a(1- N )
Network Design and Analysis-----Wang Wenjie Queueing System IV: 26
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Simple ATM Queuing System(1)
Network Design and Analysis-----Wang Wenjie Queueing System IV: 27
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Simple ATM Queuing System(2)
• Deterministic service time of one time slot
– Normalized to cell transmission time
• Always a service completion if non-empty
– Server is never idle if a job (cell) is waiting
• Simple Bernoulli is not interesting since there is no queuing
• Of interest are systems where there can be up to M > 1 arrivals per time slot, e.g., from M different input sources
Network Design and Analysis-----Wang Wenjie Queueing System IV: 28
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M-Geo/D=1/1/N Systems (1)
Network Design and Analysis-----Wang Wenjie Queueing System IV: 29
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M-Geo/D=1/1/N Systems (2)
Network Design and Analysis-----Wang Wenjie Queueing System IV: 30
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M-Geo/D=1/1/N Systems (3)
Transition Matrix
Network Design and Analysis-----Wang Wenjie Queueing System IV: 31
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Global Balance Equations
Network Design and Analysis-----Wang Wenjie Queueing System IV: 32
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Fast Computer Algorithm
Network Design and Analysis-----Wang Wenjie Queueing System IV: 33
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Blocking Probability
Network Design and Analysis-----Wang Wenjie Queueing System IV: 34
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M-Geo/D=1/1/
Network Design and Analysis-----Wang Wenjie Queueing System IV: 35
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Summary Queuing Theory(1)
Network Design and Analysis-----Wang Wenjie Queueing System IV: 36
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Summary Queuing Theory(2)
Network Design and Analysis-----Wang Wenjie Queueing System IV: 37
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Summary Queuing Theory(3)