Chapter 81 Queuing Theory Basic properties, Markovian models, Networks of queues, General service...
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Transcript of Chapter 81 Queuing Theory Basic properties, Markovian models, Networks of queues, General service...
Chapter 8 1
Queuing Theory
Basic properties, Markovian models, Networks of queues, General service time distributions, Finite
source models, Multiserver queues
Chapter 8 2
Kendall’s Notation for Queuing Systems
A/B/X/Y/Z:
A = interarrival time distribution
B = service time distribution
G = general (i.e., not specified); M = Markovian (exponential); D = deterministic
X = number of parallel service channels
Y = limit on system pop. (in queue + in service); default is
Z = queue discipline; default is FCFS (first come first served)
Others are LCFS, random, priority
Chapter 8 3
Other NotationRandom variables:
X(t) = Number of customers in the system at time t
S = Service time of an arbitrary customer
W* = Amount of time an arbitrary customer spends in the system
WQ* = Amount of time an arbitrary customer spends waiting for service
Often we are most interested in the averages or expectations: Average number in the queue
Average number in service
Q
S
L E X t
L
L
*
*Q Q
W E W
W E W
Chapter 8 4
Little’s Formula
is the time spent in the system by the nth customer. Assume these system times are uniformly finite and letbe the customer average time spent in the system.Also, let be the time average number of jobs in the system. Then under very general conditions,
where a is the arrival rate. This is usually written L = aW.The mean number of customers in the system is proportional
to the mean time in the system!
*nw
11
lim *k
k nk nw w
1
0lim ( )
s
s sX X t dt
aX w
Chapter 8 5
Heuristic Proof of Little’s Formula
Two ways to compute
Mean time in system in (0,T):
Mean number in system in (0,T):
Then
T
A(t)
D(t)
B(t) = cum. area between curves
1 0( ) * ( )
TN
nnB T w X t dt
( ) ( ) ( ) ( )W T B T N B T A T ( ) ( )X T B T T
( ) ( )lim ( ) lim lim ( ), or
( ) aT T T
B T A TX T T W T L W
A T T
Chapter 8 6
Other Little’s Formulae
In queue:
In service:
and their implications…
Expected number of busy servers
Expected number of idle servers
Single server utilization
Single server prob. of empty system
Q a QL W S aL E S
S aL E S # servers
S aL E S 1
Chapter 8 7
Observation Times
an = Proportion of arriving customers that find n in the system
dn = Proportion of departing customers that leave behind n in the system
If customers arrive one at a time and are served one at a time then
But these proportions may not match the limiting probability of having n in the system (long run proportion of time that n are in the system)
However, if arrivals follow a Poisson process then This is known as PASTA (Poisson Arrivals See Time Averages)
n na d
n na P
lim ( ) , 0,1,...n tP P X t n n
Chapter 8 8
M/M/1 Model
Single server, Poisson arrivals (rate ), exponential service times (rate )
CTMC (birth-death) model:
Chapter 8 9
M/M/1 Steady-State Probabilities
Level-crossing equations:
Solve for Pn in terms of P0
Then use the facts that
and substitute
to get
lim ( ) , 0,1,...n tP P X t n n
1, 0,1,n nP P n
0 , 1, 2,n
nP P n
1
0 01, (1 ) if 1n
nn nP r r r
/
(1 ), 0,1, if 1nnP n
Chapter 8 10
M/M/1 Performance Measures
Steady-state expected number of customers in the system
Mean time in system
Mean time in queue
Mean number in queue
0, if 1
1nnL nP
1 1 by Little's Formula
(1 )a
LW
1
=-QW W
2
=-Q QL W
Chapter 8 11
M/M/1 Performance Measures
Distribution of time in system (FCFS)
Exponential with parameter (1-)
Reversibility: Departure process is Poisson with rate
0
0
(1 )
0 1
[ * ] [ * | arrival sees customers]
(1 ) [ * | * gamma( 1, )]
(1 ) ( ) ! 1 , 0
nn
n
n
n x l x
n l n
P W x a P W x n
P W x W n
e x l e x
Chapter 8 12
Finite Capacity: M/M/1/N Model
Single server, exponential service times (rate )
Poisson arrivals (rate ) as long as there are N in the system
CTMC (birth-death) model:
Chapter 8 13
M/M/1/N Steady-State Probabilities
Level-crossing equations:
Solve for Pn in terms of P0
Then use the facts that
to get
lim ( ) , 0,1,...n tP P X t n n
1, 0,1, , 1n nP P n N
1
0 0
11, (note: need not be < 1)
1
NN n
nn n
rP r r
r
1
1, 0,1, ,
1
n
n NP n N
Chapter 8 14
M/M/1/N Performance Measures
(In the unlikely event that , for n = 1,…, N,
Steady-state expected number of customers in the system, L, has a “messy” closed form
Mean time in system
but here, a is the rate of arrival into the system
by Little's Formulaa
LW
0 1 1 )nP P N
1a NP
1 QW W
1 Q N QL P W
Chapter 8 15
Tandem Queue
If arrivals to the first server follow a Poisson process and service times are exponential, then arrivals to the second server also follow a Poisson process and the two queues behave as independent M/M/1 systems:
P{n customers at server 1 and m customers at server 2} =
1 2
1 1 2 2
1 1n m
Chapter 8 16
Open Network of Queues
• k servers, customers arrive at server k from outside the system according to a Poisson process with rate rk, independent of the other servers
• Upon completing service at server i, customer goes to server j with probability Pij, where
For j = 1,…, k, the total arrival rate to server j is
The number of customers at each server is independent and
If j < j for all j, then P{n customers at server j}
that is, each acts like an independent M/M/1 queue!
1
k
j j i iji
r P
1
n
j j
j j
1ijjP
Chapter 8 17
Closed Queuing Network
• m customers move among k servers
• Upon completing service at server i, customer goes to server j with probability Pij, where
• Let be the stationary probabilities for the Markov chain describing the sequence of servers visited by a customer:
Then the probability distribution of the number at each server is
1ijjP
1 1
, 1k k
j i ij ji j
P
1 211
, ,..., if jn
k kj
m k m jjj j
P n n n C n m
Chapter 8 18
CQN Performance
Computation of the normalizing constant Cm to get the stationary distribution can be lengthy; but may be mostly interested in the throughput
where m(j) is the arrival rate to (and departure rate from) j.
Arrival Theorem: In the CQN with m customers, the system as seen by arrivals to server j has the same distribution as the whole system when it contains only m-1 customers.
This leads to mean value analysis to find m(j) along with
Wm(j) = the average time a customer spends at server j, and
Lm(j) = the average number of customers at server j.
1
j
m mjj
Chapter 8 19
Mean Value Analysis
Solve iteratively:
11 mm
j
L jW j
, where m m m m j mL j j W j j
1
m k
i mi
m
W i
throughput
Begin with 1
1
j
W j
Chapter 8 20
M/G/1Best combination of tractability & usefulness
• Assumption of Poisson arrivals may be reasonable based on Poisson approximation to binomial distribution
– many potential customers decide independently about arriving (arrival = “success”),
- each has small probability of arriving in any particular time interval
• Probability of arrival in a small interval is approximately proportional to the length of the interval – no bulk arrivals
• Amount of time since last arrival gives no indication of amount of time until the next arrival (exponential – memoryless)
Chapter 8 21
M/G/1Best combination of tractability & usefulness
• Exponential distribution is frequently a bad model for service times– memorylessness– large probability of very short service times with occasional very
long service times
• May not want to use one of the “standard” distributions for service times, either– in a real situation, collect data on service times and fit an empirical
distribution
• Distributions of number of customers in the system and waiting time depend on service time distribution to be specified
Chapter 8 22
M/G/1Best combination of tractability & usefulness
• Assumption of Poisson arrivals may be reasonable based on Poisson approximation to binomial distribution
– many potential customers decide independently about arriving (arrival = “success”),
- each has small probability of arriving in any particular time interval
- Distributions of number of customers in the system and waiting time depend on service time distribution
Chapter 8 23
M/G/1 PerformanceHow many customers? How much time?
S is the length of an arbitrary service time (random variable)
is the arrival rate of customers; define = E[S] and assume it is < 1.
Expected values can be found from generalizing Little’s formula from # customers in the system to amount of work in the system:
An arriving customer brings S time units of work:
The time average amount of work in the system (V)
= * the customer average amount of work remaining in the system
Chapter 8 24
Work ContentWQ* is the (random variable) waiting time in queue
Expected amount of work per customer is
If a customer’s service time
is independent of own wait in queue, get average work in system
*
0
2
*
2
S
Q
Q
E SW S x dx
E SE SW
Workremaining
S
Begin service
Enter Depart
2
*
2Q
E SV E S E W
Chapter 8 25
Mean waiting time
WQ = Customer mean waiting time = average work in the system
when a customer arrives
From PASTA, WQ = V. Therefore, (Pollaczek-Khintchine formula)
And the other measures of performance are:
2 2
2 2 1Q Q Q
E S E SW E S W W
E S
2 2
, ,2 1Q Q Q
E SL W W W E S L W
E S
Chapter 8 26
Priority Queues
Different types of customers may differ in importance.
• Type i customers arrive according to a Poisson process with rate i and require service times with distribution Gi, i = 1, 2.
• Type 1 customers have (nonpreemptive) priority: – service does not begin on a type 2 customer if there is a type 1
customer waiting.
– If a type 1 customer arrives during a type 2 service, the service is continued to completion.
What is the average wait in queue of a type i customer, i
QW
Chapter 8 27
Two customer types w/o priority
M/G/1 model with
Average work in system is
If the server is not allowed to be idle when the system is not empty, this quantity is the same for the system with priority.
1 21 2 1 2G x G x G x
2 221 1 2 2
1 1 2 2
2 21 1 2 2
1 1 2 2
2 1 2 1
2 1
E S E SE SV
E S E S E S
E S E S
E S E S
Chapter 8 28
Two customer types with priority
Let Vi be the average amount of type i work in the system
Now focus on a type 1 customer. Waiting time = amt. of type 1 work in system + amt. of type 2 work in service when this customer arrives, so
2
2
i ii ii i Q
E SV E S W
in queue in servicei
QV iSV
2 2
1 1 2 21 1 2 11 1 2 2Q S Q
E S E SW V V E S W
Chapter 8 29
Two customer types with priority
2 2
1 1 2 211 1
1 1
if 12 1Q
E S E SW E S
E S
But a type 2 customer has to wait for everyone ahead, plus any type 1 customers who arrive during the type 2 wait, so
2 21 1 2 22 2 2
1 11 1 2 2 1 1
1 1 2 2
2 1 1
if 1
Q Q Q
E S E SW V E S W W
E S E S E S
E S E S
Chapter 8 30
M/M/k Model
k identical machines in parallel, Poisson arrivals (rate ), exponential service times (rate )
CTMC (birth-death) model:
Chapter 8 31
M/M/k Steady-State Probabilities
Level-crossing equations:
Define =/k, solve for Pn in terms of P0
Then use the facts that
to get
1
1
1 , 0,1,..., 1
, , 1,...n n
n n
P n P n k
P k P n k k
( )0!
0!
, 0,1,...,
, 1, 2,...
n
k n
kn
n kk
P n kP
P n k k
1
0 01, (1 ) if 1n
nn nP r r r
1
1
0 0
( ) ( ) if 1
! (1 ) !
n kk
n
k kP
n k
is the utilizationof each server
Chapter 8 32
M/M/k Performance Measures
Steady-state expected number of customers in the system
Mean flow time
Expected waiting time
Expected number in the queue (k is expected number of busy servers)
020
( ), if 1
! (1 )
k
nn
kL nP k P
k
0
1 1 ( )= by Little's Formula
- !(1 )
kL kW P
k k
1QW W
Q QL W L k
Chapter 8 33
Erlang Loss System
M/M/k/k system: k servers and a capacity of k: an arrival who finds all servers busy does not enter the system (is lost)
Above is called Erlang’s loss formula, and it holds for M/G/k/k as well, if
0
( ), 0,1,...,
!
n
n
kP P n k
n
0
( ) ( ), 0,...,
! !
i nk
i n
k kP i k
i n
k E S