M/M/1 queue
description
Transcript of M/M/1 queue
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M/M/1 queue
λn = λ, (n >=0); μn = μ (n>=1)
λ: arrival rateμ: service rate
λ μ
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Traffic intensity rho = λ/μ
It is a measure of the total arrival traffic to the system Also known as offered load
Example: λ = 3/hour; 1/μ=15 min = 0.25 h
Represents the fraction of time a server is busy In which case it is called the utilization factor
Example: rho = 0.75 = % busy
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Queuing systems: stability λ<μ
=> stable system
λ>μ Steady build up of customers => unstable
Time 1 2 3 4 5 6 7 8 9 10 11
123
busy idleN(t)
Time 1 2 3 4 5 6 7 8 9 10 11
123
N(t)
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Example#1 A communication channel operating at 9600 bps
Receives two type of packet streams from a gateway Type A packets have a fixed length format of 48 bits
Type B packets have an exponentially distribution length With a mean of 480 bits
If on the average there are 20% type A packets and 80% type B packets
Calculate the utilization of this channel Assuming the combined arrival rate is 15 packets/s
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Performance measures L
Mean # customers in the whole system
Lq
Mean queue length in the queue space
W Mean waiting time in the system
Wq
Mean waiting time in the queue
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Mean queue length (M/M/1)
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Mean queue length (M/M/1) (cont’d)
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Little’s theorem This result
Existed as an empirical rule for many years And was first proved in a formal way by Little in 1961
The theorem Relates the average number of customers L
In a steady state queuing system
To the product of the average arrival rate (λ) And average waiting time (W) a customer spend in a system
WL .
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LITTLE’s Formula
: average number of messages in system : average delay λ: arrival rate
Little’s relation holds for any Service discipline Arrival process Holding area
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Graphical Proof A(t)
Cumulative arrival process L(t)
Nb. of customers that left system up to t
=> N(t) = A(t) – L(t) Nb. of customers in system at time t
di : interval between ith arrival and its departure
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Graphical Proof (continued)
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Graphical Proof (continued)
Now, let
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Mean waiting time (M/M/1) Applying Little’s theorem
1
1.1
.LW
WL
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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
ii zPzg
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M/M/1 Queue – Infinite Waiting Room Probability generating function
Mean
Variance
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M/M/S
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M/M/S (cont’d)
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M/M/S
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λ
μS servers
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M/M/S: normalizing equations
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M/M/S: stable queue is λ/Sμ < 1 ?
Otherwise you will not get a stable queue, as such
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M/M/S: performance measures Mean queue length
Mean waiting time in the queue (Little’s theorem)
Mean waiting time in the system
Mean # of customers in the whole system
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Erlang C formula A quantity of interest
Probability to find all s servers busy
Ratio between Lq and Pc
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M/M/S: stability revisited Stable
If λ/Sμ < 1
Arrival rate to an individual server
Utilization of a server
Utilization of all servers
S
1.S
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M/M/1/N
Birth and death equations
λ μ% loss
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M/M/1/N: normalizing constant Let ρ=λ/μ
As such
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Probability of arrivingto a full waiting room
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M/M/1/N: what percent of λ gets into the queue? Percentage of time the queue is full
is equal to PN
Rate of lost customers = λ.PN
Rate of customers getting in : λ.(1-PN) Often referred to as effective customer arrival rate
Utilization of server
.75 .25
fullNot full
)1.( NP
)1.( NP
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M/M/1/N: performance measures Mean # of customers in the system
Mean queue length
Waiting time in system: W = L/λ
Waiting time in queue: Wq = Lq/λ
1
1
1)1(
1
N
NNL
LM/M./1
)1( 0PLLq
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M/M/1/N: equivalent systems When an M/M/1/N queue is full
Continuous arrival A system with loss
is equivalent to shutting up the service For the duration during which the queue is full
And starting it up again when system no longer ful
This system is called a shut down system
This equivalence holds only when the inter-arrival is exponential
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Proof: rate diagrams M/M/1/N system with loss
Consider the special case where N = 5
0 1 2 3 4 5λ λ λ λ λ
μ μ μ μ μ
λ
45545
201
10
....).(..
..).(..
PPPPP
PPPPP
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Proof: rate diagrams (cont’d) M/M/1/N shut down system
Consider the special case where N = 5
0 1 2 3 4 5λ λ λ λ λ
μ μ μ μ μ
45
201
10
....
..).(..
PP
PPPPP
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M/M/infinity: birth and death equations
.
.
λ μ
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1....
...
.
Pn
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Pn
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Infinite number ofservers
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M/M/infinity: normalizing constant
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Erlang system: M/M/S/S
.
.
λ μ
Finite number ofServers = S
0
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2
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1
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...!2!1
1
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n
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Erlang loss formula What percent gets in and
What percent gets lost
PS = prob S customers in system
Effective arrival rate
Rate of lost customers = λ.PS
)1.( SP
S
n
n
S
S
n
SP
0 !
!/
Erlang loss formula
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Erlang B formula Probability of finding all s servers busy
In an iterative form: