Pertemuan 19 s/d 22
(minggu ke-10 dan ke-11)
Chapter 8 1

Basic properties, Markovian models, Networks of queues, General service time distributions, Finite source models, Multiserver queues
Chapter 8 2

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

Random 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
W
Q
* = Amount of time an arbitrary customer spends waiting for service
Often we are most interested in the averages or expectations:
L 
 
W
  
L
Q

Average number in the queue
W
Q

Q
*

L
S

Average number in service
Chapter 8 4

w n
* is the time spent in the system by the n th customer.
Assume these system times are uniformly finite and let w
 lim k

1 k
 k n

1 be the customer average time spent in the system.
s
Also, let X
 lim s

1 s
0 be the time average number of jobs in the system.
Then under very general conditions, where l a

X
 l a w is the arrival rate. This is usually written L = l a
W .
w n
*
The mean number of customers in the system is proportional to the mean time in the system!
Chapter 8 5

A ( t )
B ( t ) = cum. area between curves
D ( t )
Two ways to compute
Mean time in system in (0, T ):

T
 N w
  T n

1
 n
*
0
( ) ( ) N

( ) ( )
Mean number in system in (0, T ): ( )

( )
Then
T lim

X T

T lim

( ) ( )
( ) T

T lim
 l  
( ), or L
 l a
W
Chapter 8 6

In queue: L
Q
In service: L
S

 l a
W
Q l a
  and their implications…
Expected number of busy servers L
S
 l a

Expected number of idle servers
Single server utilization
Single server prob. of empty system
L
S
# servers

1 l
 a

 





Chapter 8 7

a n d n
P n
 lim t


( )
 
,

0,1,...
= Proportion of arriving customers that find n in the system
= 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 a n
 d n
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 a n

P n
This is known as PASTA (Poisson Arrivals See Time Averages)
Chapter 8 8

Single server, Poisson arrivals (rate l
) , exponential service times (rate

)
CTMC (birth-death) model:
Chapter 8 9

P n
 lim t


( )
 
,

0,1,...
Level-crossing equations: l
P n
 
P n

1
, n

0,1,
Solve for P n in terms of P
0
P n

  n
0
,

1, 2,
Then use the facts that
  n

0
P n

1,
  n

0 r n   r

1
) if r

1 and substitute to get
P n
  n
(1
 
), n

0,1, if
 
1
Chapter 8 10

Steady-state expected number of customers in the system
L
Mean time in system
   n

0 nP n

1



, if
 
1
W
 l
L a


(1
1
 
)

1
 l
Mean time in queue
W
Q

W

1

=  l
by Little's Formula

Mean number in queue
L
Q
 l
W
Q
= l
2
 
Chapter 8 11

Distribution of time in system (FCFS)
[ *
 x ]


  n

0 n
[ *
 x n
  n

0
(1
   n
P W
 x W * gamma( n
 
   n

0
(1
   n
  l n 1 e
  x
(
 x ) l l
  e
 
(1
 
) x
, x

0
Exponential with parameter

(1-

)
Reversibility: Departure process is Poisson with rate l
Chapter 8 12

N
Single server, exponential service times (rate

)
Poisson arrivals (rate l
) as long as there are

N in the system
CTMC (birth-death) model:
Chapter 8 13

N
P n
 lim t


( )
 
,

Level-crossing equations: l
P n
 
P n

1
, n

0,1, , N

1
Solve for P n in terms of P
0
0,1,...
Then use the facts that to get P n


 n

0
P n

1,

N n

0 r n 
1
 r
N

1
1
 r
 l   
1

1
 l 
 l  
N

1

, n

0,1, ,
Chapter 8
N
14

N
(In the unlikely event that l  
, for n
= 1,…,
N , P n

P
0

1

N
 
Steady-state expected number of customers in the system, L , has a “messy” closed form
Mean time in system W

L l a
by Little's Formula but here, l a is the rate of arrival into the system l a
 l 
1

P
N

W
Q

W

1

L
Q
 l 
1

P W
N

Q
Chapter 8 15

l

1

2
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} =

 l

1
 
 
1
 l

1

 l

2
 
 
1
 l

2


Chapter 8 16

• k servers, customers arrive at server k from outside the system according to a Poisson process with rate r k
, independent of the other servers
• Upon completing service at server i , customer goes to server
For j j with probability
= 1,…,
P ij
, where
 j
P
 ij
1 l  i

1
The number of customers at each server is independent and
If l j
<
 j k , the total arrival rate to server for all j , then P{ n j is customers at server j }  j
  j

 l
 j j
 
  1 k
 l
 l
P i ij j j

 that is, each acts like an independent M/M/1 queue!
Chapter 8 17

• m customers move among k servers
• Upon completing service at server i , customer goes to server j with probability P ij
, where
 j
P
 ij
1
• Let p be the stationary probabilities for the Markov chain describing the sequence of servers visited by a customer: p j
 i k

1 p
P i ij k 
,
 j

1 p j

1
Then the probability distribution of the number at each server is m

,
1 2
,..., n k
 
C m j k 

1


 p
 j j


 n j
if j k 

1 n j
 m
Chapter 8 18

Computation of the normalizing constant C m to get the stationary distribution can be lengthy; but may be mostly interested in the throughput where l l m
  j j

1 l m
  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 l m
( j ) along with
W m
( j ) = the average time a customer spends at server j , and
L m
( j ) = the average number of customers at server j .
Chapter 8 19

Solve iteratively:
W m
  
1

L m

1
 
 j
L m
   l m
   
, where l m l m
 m
 i k

1 p
W i m
 
Begin with W
1
  

1 j
 p l j m throughput
Chapter 8 20

Best 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

Best 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

Best 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

S is the length of an arbitrary service time (random variable) l is the arrival rate of customers; define

= l
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 )
= l
* the customer average amount of work remaining in the system
Chapter 8 24

W
Q
* is the (random variable) waiting time in queue
Expected amount of work per customer is


*
Q
 
0
S 
S
  

Work remaining
S


*
Q


2
If a customer’s service time Enter
Begin service
Depart is independent of own wait in queue, get average work in system l
V
 l  
*
Q


2
Chapter 8 25

W
Q
= Customer mean waiting time = average work in the system when a customer arrives
From PASTA, W
Q
= V . Therefore, (Pollaczek-Khintchine formula)
W
Q
 l  
Q
 l
2

W
Q
 l
 l

And the other measures of performance are:
L
Q
 l
W
Q
 l
2
 l
 , W

W
Q
  
, L
 l
W
Chapter 8 26

Different types of customers may differ in importance.
• Type i customers arrive according to a Poisson process with rate l i and require service times with distribution G i
, 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, W
Q i
Chapter 8 27

M/G/1 model with l l l
1 2
 l
1 l
G x
1
   l
2 l
G
2
 
Average work in system is
V
 l
 l





1



1
2


1

  l l
2
 l l
2

2
2

 
     



 l
1


1
2
 l
1

  l
2
 l
2

2
2

  
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.
Chapter 8 28

Let V i be the average amount of type i work in the system
V i  l i
  i i
Q in queue
 l i
E S i
2
 
2 in service
V
Q i
V
S i
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
W
1 
V
1
Q

V
S
2  l
1
 
1
Q
 l
1

1
2

2
 l
2

2
2

2
Chapter 8 29

W
Q
1  l
1

1
2


 l
1
 l
2
 
1

2
2

 if l
1
 
1

1
But a type 2 customer has to wait for everyone ahead, plus any type 1 customers who arrive during the type 2 wait, so
W
Q
2   l
1
 
Q
2 
W
Q
2 
 l
1

1
2
  l
2 l
1
   l
2
  

2
2


1
 l
1 if l
1
 
1
 l
2
 
2

1

Chapter 8 30

k
k identical machines in parallel, Poisson arrivals (rate l
) , exponential service times (rate

)
CTMC (birth-death) model:
Chapter 8 31

k
Level-crossing equations: l
P n
  n

1
 
P n

1
, n

0,1,..., k

1 l
P n
  n

1
, n

,

1,...
Define

= l
/k
P n

, solve for

 

( k

) n k n !
k  n k !
P n
0
,

0,1,...,
P n
0
,
 
1, k

2,...
Then use the facts that

P
 n n

0 in terms of
P n
 k
1,
 
P n

0 r
0 n   r

1
) if r

1 to get
P
0
 

 k

1 n

0
( k n

) n
( k


) k
!

(1

) !


1
if
 
1
 is the utilization of each server
Chapter 8 32

k
Steady-state expected number of customers in the system
L
   n

0 nP n
 k
 



( k k

!
) k

 (1



)
2





P
0
, if
 
1
Mean flow time
W

L 1 l 


 k
1
 l




( k

) k k !(1
 
)


P
0
by Little's Formula
Expected waiting time
W
Q

W

1

Expected number in the queue ( k
 is expected number of busy servers)
L
Q
 l
W
Q
  k

Chapter 8 33

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)
P n

( k

) n n !
,

0,1,..., k
P i

( k

) i i !
 k n

0
( k

) n n !
, i

0,..., k
Above is called Erlang’s loss formula, and it holds for M/G/ k / k as well, if k
 l  
Chapter 8 34