basic queueing theory (i)
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Cheng-Fu Chou, CMLab, CSIE, NTUCheng-Fu Chou, CMLab, CSIE, NTU
Basic Queueing Theory (I)
Cheng-Fu Chou
P. 2
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Outline
Little resultM/M/1 Its variantMethod of stages
P. 3
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Queueing System
Kendall’s notations– A/B/C/K– C: number of servers– K: the size of the system capacity; the buffer
space including the servers A(t): the inter-arrival time dist. B(t): the service time dist.
– M: exponential dist.– G: general dist.– D: deterministic dist.
P. 4
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Time Diagram for queues
Cn: the n-th customer to enter the systsem N(t): number of customers in the system at time t U(t): unfinished work in the system at time t n: arrival time for Cn
tn: inter-arrival time between Cn-1 and Cn, i.e., A(t) = P[tn t]
xn: service time for Cn, B(t) = P[xn t]wn: waiting time for Cn
sn: system time for Cn= wn+xn
– Draw the diagram
P. 5
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Little Result
(t): no. of arrivals in (0,t) (t): no. of departures in (0,t) t: the average arrival rate during the interval (0,t) r(t): the total time all customers have spent in the
system during (0,t) Tt: the average system time during (0,t)
– proof
P. 6
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/1
The average inter-arrival time is t = 1/ and t is exponentially distributed.
The average service time is x = 1/ and x is exponentially distributed.
Find out – pk : the prob. of finding k customers in the
system – N : the avg. number of customers in the system– T : the avg. time spent in the system
P. 7
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/1
Poisson arrival
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Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Discouraged Arrival
A system where arrivals tend to get discouraged when more and more people are present in the system– arrival rate: k = /(k+1) , where k = 0,1,2,…
– service rate: k = , where k = 1,2,3,…
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Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Discouraged Arrival
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Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/
Infinite number of servers– there is always a new server available for each
arriving customer.– arrival rate : – service rate of each server:
P. 11
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/
We know– Arrival rate k = , k = 0, 1, 2, …
– Departure rate k = k , k = 1, 2, 3, …
P. 12
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/m
The m-server case– The system provides a maximum of m servers
P. 13
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/m
Arrival rate k = and service rate k = min(k, m)
P. 14
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/1/K
Finite storage: a system in which there is a maximum number of customers that may be stored ( K customers)
P. 15
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/1/K
P. 16
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/m/m
m-server loss system
P. 17
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/m/m (m-server loss system)
m-server loss systems
P. 18
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/1//m
Finite customer population and single server– A single server– There are total m customers
P. 19
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/M/1//m (finite customer population)
P. 20
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
PASTA
Poisson Arrival See Time Average
P. 21
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Method of stages
Erlangian distribution
P. 22
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Er: r-stage Erlangian Dist.
r-stage Erlangian dist.
P. 23
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/Er/1
P. 24
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
E2/M/1
P. 25
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Bulk arrival systems
Bulk arrival system
– gi = P[bulk size is i]
– e.g. random-size families arriving at the doctor’s office for individual specific service
P. 26
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Bulk Service System
Bulk service system– The server will accept r customers for bulk
service if they are available– If not, the server accept less than r customers if
any are available
– HW : M/B2/1
P. 27
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
M/B2/1
P. 28
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Response time in M/M/1
The distribution of number of customers in systems :
How about the distribution of the system time ?– Idea: if an arrival who
finds n other customers in system, then how much time does he need to spend to finish service?
,...1,0,)1( ip ii
1
1
n
jjn TT
P. 29
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Response time (cont.)
rn: the proportion of arrivals who find n other customers in system on arrival
pn: the proportion of time there are n customers in system
Due to PASTA, {rn} = {pn}, given that there are n customers in the systems
1
}|{
n
sT
sneE
P. 30
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Response Time
Unconditioning on n
sss
spneEeE
n
n
n
n
n
nn
sTsT
0
0
1
0
)1(}|{}{
)exp( T
P. 31
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
Waiting time Dist. For M/M/c
For M/M/c queueing system, given a customer is queued, please find out his/her waiting time dist. is – (D| D>0) ~ exp(c – )– hint
11
00
0
0
!
)(
!)1(
)(
cnfor !/
for !/)(
c
j
jc
cn
n
n
j
c
c
cp
ccp
cnncpp
P. 32
Cheng-Fu Chou, CMLAB, CSIE, NTUCheng-Fu Chou, CMLAB, CSIE, NTU
W = P(D>0)/(c-) And
1
0
1)0(c
jjpDP
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