network design and analysis-----wang wenjie queueing system iv: 1 © graduate university, chinese...

Network Design and Analysis-----Wang Wenjie Queueing System IV: 1

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Network Design and Analysis

Wang Wenjie

[email protected]


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Queueing System IV

Discrete-Time Queuing Systems


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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


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1.1 Motivation

Some queuing systems operate on a slotted

time basis: DT Systems


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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)


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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-


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Transition Probability Matrix


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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


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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


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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.


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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.


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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


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Geometric Distribution (2/2)

• Geometric distribution has the memoryless property

• Mean : 1/p

• Variance : (1-p)/p2


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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


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1.3. Geo/Geo/1 Systems

• Bernoulli arrivals, geometric service times

(DT analog to M/M/1)


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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.


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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


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State Probabilities

0prp nn

rpr

prpp

n

n

nn

111 0

0

000

nn rrp )1(


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Average # of Customers

If 0< r <1 then:

00

)1(][n

n

n

n nrrnpnE

r

rnE

1][


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Traffic Intensity

s

aar

apserveridlep

1)1)(1(

)1(][ 0

s

a


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Mean Delay

)1(

][][

ra

r

a

nEE

• System throughput (customers/slot time) = a =

• Apply Little’s law


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1.4. Geo/Geo/1/N Systems

• Similar to Geo/Geo/1, except for state N


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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


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State Probabilities

1

0

0)1(

1N

n

Nn arrp

)1(1

110 rarr

rp

NN


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Delay

• P[block] = P[N in system] = N

• So, throughput is a(1- N )


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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: 1 © graduate university, chinese...

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