M/G/1 variants and Priority Queue
Cheng-Fu Chou
Cheng-Fu Chou, CMLab, CSIE, NTU
HW. 1 M/G/1 with bulk service
 Consider an M/G/1 system with bulk service.
Whenever the server becomes free, he accepts 2
customers from the queue into service simultaneously,
or , if only one is on queue, he accepts that one; in
either case, the service time for the group (of size 1
or 2) is taken from B(x). Let qn be the number of
customers remaining after the nth service instant. Let
vn be the number arrivals during the nth service.
Define B*(s), Q(z), and V(z) as transforms associated
with the random variables x, q, v as usual. Let r= lx/2
P. 2
Cheng-Fu Chou, CMLAB, CSIE, NTU
– Using the method of imbedded Markov chain, find
E(q) in term of r, P(q=0) =p0.
– Find Q(z) in term of B*(.), p0, p1= P(q=1)
– Express p1 in terms of p0
Ans:
(a) E[q]= r + (2(1-p0)+l2E[x2]-4r2)/(4(1-r))
(b) Q(z)= B*(l-lz) (p0(1-z2)+p1z(1-z))/(B*(l-lz)-z2)
(c) p1 = 2(1-p0-r)
P. 3
Cheng-Fu Chou, CMLAB, CSIE, NTU
(a) Clearly wecan write
 qn  2  vn 1 qn  2

qn 1   qn  1  vn 1 qn  1
v
qn  0
 n 1
Introducing thefunction
 2 k2
 2, k  
k 0  k  2
We haveq n 1  q n -  2,q n  v n 1 , lettingn   and
takingexpection
q  q - E[ 2,q ]  v
But

E[ 2,q ]    2,k P[q  k ]
k 0

 P [q  1]   2P [q  k]
k 2
 p1  2(1- p 0 - p1 )
So, v  E[ 2,q ]  2 - 2p0 - p1
P. 4
Cheng-Fu Chou, CMLAB, CSIE, NTU
HW2 M/G/1 (service time)
 Consider an M/G/1 queueing system in which service is
given as follows. Upon entry into service, a coin is
tossed, which has probability p of giving Heads. If the
result is Heads, the service time for that customer is
0 seconds. If Tails, his service time is drawn from the
following uniform distribution:
f(x)=1/(b-a), if a<x<b; otherwise f(x)=0
– Find the average service time x
– Find the variance of service time
– Find the expected waiting time
– Find W*(s)
P. 5
Cheng-Fu Chou, CMLAB, CSIE, NTU
M/G/1 with vacations
 Consider a first-come-first-served M/G/1 queue with
the following changes. The server serves the queue as
long as someone is in the system. Whenever the system
empties the server goes away on vacation for a certain
length of time, which may be a random variable. At the
end of his vacation the server returns and begins to
serve customers again: if he returns to an empty system
then he goes away on vacation. Let

– F ( z) 
f j z j be the z-transform for the number
of customers awaiting service
j 1
when the server returns from vacation to find at
least one customer waiting

P. 6
Cheng-Fu Chou, CMLAB, CSIE, NTU
(a) Derive an expression which gives qn+1 in term of qn,
vn+1 and j
(b) Derive Q(z) in term of p0
(c) Show that p0 = (1-r)/F1(1), r = lx
(d) Assume now that the service vacation will end
whenever a new customer enters the empty system.
For this case find F(z) and show that when we
substitute it back into our answer for (b) then we
arrive at the classical M/G/1 solution.
P. 7
Cheng-Fu Chou, CMLAB, CSIE, NTU
(a) Clearly,if q n  0, q n 1  q n  1  v n 1
as for theusual M/G/1 system
If q n  0 theservergoes on vacation and does
not begin servinguntil thereare f  1 in thesystem.
T hus,q n 1  f - 1  vn 1 for q n  0.
Let  f,k
k k  0

f k  0
So,
q n 1   f,k  1  v n 1
P. 8
Cheng-Fu Chou, CMLAB, CSIE, NTU
(b) We have
 f, qn -1 v n 1
Q n 1 (z)  E[zq n1 ]  E[z
]

 E[zv n1 1 ]E[z f, qn ]
by indepenedence of vn 1 , f, q n . Lettingn   we find
Q(z) 
and E[z
 f, q
V(z)

E[z f, q ]
z

]  E[z ]p[q  0]   z k p[q  k ]
f
k 1
 F(z)p0  [Q(z)- p 0 ]
p 0 (1- F(z))
Q(z)  V(z)
and V(z)  B * (l - lz)
V(z) - z
p (1- F(z))
Q(z)  B * (l - lz) 0
B * (l - lz) - z
Cheng-Fu Chou, CMLAB, CSIE, NTU
P. 9
(c) From (b), we have
p (1- F(z))
Q(z)  V(z) 0
V(z) - z
T o determinep 0 , we evaluatetheabove eqn. at z  1
(using L' Hospitalrule).T hen,
p 0 (1- F(z))
Q(z)
1
 lim
V(z) z 1 B * (l - lz) - z
p 0 (-F(1) (1))
p 0 F(1) (1)


(1)
B * (0)(-l ) - 1
1- l x
1- r
p 0  (1)
F (1)
Cheng-Fu Chou, CMLAB, CSIE, NTU
P. 10
(d) In t hiscase, f1  1 and f k  0 for k  1.
So, F(z)  z and F(1) (1)  1, p 0  1 - r .
(1- r )(1- z)
Q(z)  B * (l - lz)
B * (l - lz) - z
which is t heP - K t ransform equat ion for M/G/1
P. 11
Cheng-Fu Chou, CMLAB, CSIE, NTU
M/G/1 with vacations
 At the end of each busy period, the server goes on
vacation for some random interval of time
 A new arrival to an idle system rather than going into
service immediately, waits for the end of the vacation
x1
x2
x3 x4 v1 v2 v3 x5
P. 12
Cheng-Fu Chou, CMLAB, CSIE, NTU
 Let v1, v2, …, be the duration of successive vacations
taken by the server and they are i.i.d. r.v.
 Observation
– A new arrival to the system, waits for the
completion of the current service or vacation
o So, the waiting time formula W = R/(1-r) is still
valid
– R is the mean residual time for completion of the
service or vacation
P. 13
Cheng-Fu Chou, CMLAB, CSIE, NTU
 By using the same graphical argument
rt
x1
x1
x2
time t
v1
v2
xM(t)
P. 14
Cheng-Fu Chou, CMLAB, CSIE, NTU
 Residual service time for an
M/G/1 system with vacations
– M(t): # of services
completed by time t
– L(t) : # of vacations
completed by time t
t
1
1M (t ) 1 2 1 L (t ) 1 2
R   r (t )dt   xi   vi
t0
t i 1 2
t i 1 2
M (t )
L (t )
1 2
1 2
xi
vi


M(t) i 1 2
L(t ) i 1 2


*
t
M(t)
t
L(t )
P. 15
Cheng-Fu Chou, CMLAB, CSIE, NTU
T akinglimit as t  
M(t)
l
t
L(t) 1  r
t(1- r )

,
v
t
L(t)
v
λ x 2 ( 1  ρ)v 2
R  * 

2
2v
R
λx2
v2
W 


1  r 2(1  r ) 2v
P. 16
Cheng-Fu Chou, CMLAB, CSIE, NTU
M/G/1 with feedback queue
 Consider an M/G/1 system in which a departing
customer immediately joins the queue again with
probability p, or departs forever with probability q =
1- p. Service is FCFS, and the service time for a
returning customer is independent of his previous
service time. Let B*(s) be the transform for the
service time pdf and let B*T(s) be the transform for a
customer’s total service time pdf.
(a) Find B*T(s) in term of B*(s), p and q
(b) Find QT(z)
(c) Find N, the average number of customer in the
systsem
P. 17
Cheng-Fu Chou, CMLAB, CSIE, NTU
(a) Conditioning on thenumber of return t rips, we have
B*T ( s | exactlyn return t rips)  (B* (s))n 1
Unconditioningyields

B ( s )   BT* ( s | exactlyn return t rips)p(exactly n rest urn trips)
*
T
n 0


n 0
n 0
  (B* (s))n 1 qpn  qB* ( s ) [ pB* ( s )]n
P. 18
Cheng-Fu Chou, CMLAB, CSIE, NTU
(b) In determining the number in the system, we may
assume that a customer cycles backs directly into
service instead of to the tail of the queue. This is
allowed due to the “memoryless” selection of a new
service time each time a customer returns in addition
to the independence of the feedback decision. Thus,
we may consider our queue as an M/G/1 system with
B*T(s) as the transform for the service time.
P. 19
Cheng-Fu Chou, CMLAB, CSIE, NTU
(1  rT )(1  z )
QT ( z )  B (l  lz ) *
BT (l  lz )  z
*
T
*
qB
(s)
lx
*
whereBT 
and rT 
*
1  pB ( s)
q
P. 20
Cheng-Fu Chou, CMLAB, CSIE, NTU
(c) For M/G/1, N  r 
l x
2
2
2(1  r )
.
l xT
So, N  rT 
2(1  rT )
2
2
P. 21
Cheng-Fu Chou, CMLAB, CSIE, NTU
Priority Queue
 M/G/1 system with n different priority classes
– class 1 > class 2 > class 3 >…
– Arrival rate: lk
– Mean service time: xk = 1/mk
– Second moment of service time: xi 2
P. 22
Cheng-Fu Chou, CMLAB, CSIE, NTU
Nonpreemptive priority
 A customer undergoing service is allowed to complete
service without interruption even if a customer of
higher priority arrives in the meantime. A separate
queue is maintained for each priority class
– Goal: find an equation for average delay for each
priority class
– Total n classes
– NQk: average number on queue for class k
– Wk: average queueing time for class k
– rk = lk/mk: system utilization for class k
– R: mean residual service time
P. 23
Cheng-Fu Chou, CMLAB, CSIE, NTU
1
 Assume that
 W1  R 
N Q1
μ1
overall system
utilization is less
By Lit tle's result
than 1, i.e., r1  r2
...  rn < 1
N Q1  λ1W1,W1 
R
1  ρ1
similarly,
W2  R 
W2 
1 1 1 2 1
NQ 
N Q  λ1W2
μ1
μ2
μ1
R  ρ1W1
1-ρ1-ρ2
R
W2 
( 1-ρ1 )(1-ρ1-ρ2 )
Cheng-Fu Chou, CMLAB, CSIE, NTU
P. 24
T hederivationis similar for all priorityclasses k  1
Wk 
R
(1- r1 - r 2 - ...- r k -1 )(1- r1 - ...- r k )
 T heavg. delay per customerof class k is
1
Tk 
 wk
mk
Moreover,themean residual service timeR
1 n
R   li xi2
2 i 1
n
 Wk 
2
l
x
 i i
i 1
2(1- r1 - r 2 - ...- r k -1 )(1- r1 - ...- r k )
Cheng-Fu Chou, CMLAB, CSIE, NTU
P. 25
HW
 Consider a nonpreemptive system and 2 customer
classes A and B with respective arrival and service
rate lA, mA, and lB, mB .
If mA > mB show that the average delay per customer
(avg. over both classes)
T = lATA + lBTB/(lA+lB) is smaller when class A with
higher priority (class A > class B) than the case when
class B with higher priority ( class B > class A)
P. 26
Cheng-Fu Chou, CMLAB, CSIE, NTU
Preemptive resume priority
 Service of a customer is interrupted when a higher
priority customer arrives and is resumed from the
point of interruption once all customers of high
priority have been served.
P. 27
Cheng-Fu Chou, CMLAB, CSIE, NTU
1.
1
mk
k
2.
Rk
, where R k 
1 - r1 - ...- r k
2
l
x
 i i
i 1
2
Rk
 k 1 
3. Tk 

   r i Tk
m k 1 - r1 - ...- r k  i 1 
1
1
T1 
m1
1  r1   R1
1  r1
1
,
1  r1  ......- r k   Rk
mk
Tk 
1  r1  ......- r k -1 1  r1  ......- r k 
P. 28
Cheng-Fu Chou, CMLAB, CSIE, NTU
CASE Study
 DTMC for Slotted Aloha
 CTMC for Wireless handoff model
 DTMC for 802.11 Model
P. 29
Cheng-Fu Chou, CMLAB, CSIE, NTU
Slotted Aloha Model
 The Aloha network was developed to provide radiobased data communication.
– Slotted system
– Collision or perfect reception
– Consider m users, n of which are currently
backlogged. Each of the m-n unbacklogged users is
assumed to transmit independently in each slot
with probability a, while each backlogged user
transmits independently in each slot with
probability b
P. 30
Cheng-Fu Chou, CMLAB, CSIE, NTU
DTMC for Slotted Aloha
Let thenumber of backlogged users, n, denotethestateof theDT MC.
Let A(i,n) denotetheprobability thati unbacklogged users attempt totransmit
in a slot whentheDT MCis in staten and B(i, n) be thecorresponding probability
for backlogged users.
m  n
1  a m n i a i , 0  i  m -n
A(i, n)  
 i 
n
n i
B(i, n)   1  b  b i , 0  i  n
i
P. 31
Cheng-Fu Chou, CMLAB, CSIE, NTU
DTMC for Slotted Aloha
P. 32
Cheng-Fu Chou, CMLAB, CSIE, NTU
Wireless Handoff Model
 Consider the performance model of a single cell in a
cellular wireless communication network.
– New calls arrive in a Poisson stream at the rate l1
– Handoff calls arrive in a Poisson stream at the rate
l2
– An ongoing call (new or handoff) completes service
at the rate m1
– The mobile engaged in the call departs the cell at
the rate m2
P. 33
Cheng-Fu Chou, CMLAB, CSIE, NTU
More…
 There are a limited number of channels, n, in the
channel pool.
 When a handoff call arrives and an idle channel is
available in the channel pool, the call is accepted and a
channel is assigned to it. Otherwise, the handoff call
is dropped.
 When a new call arrives, it is accepted provided g+1 or
more channel are available in the channel pool;
otherwise the new call is blocked. Here, g is the
number of guard channels used to give priority to
handoff calls.
P. 34
Cheng-Fu Chou, CMLAB, CSIE, NTU
CTMC for Handoff Model
P. 35
Cheng-Fu Chou, CMLAB, CSIE, NTU
DTMC for Memory Interference in
Multiprocessor System
 The processors’ ability to share the entire memory
space provides a convenient means of sharing
information and provides flexibility in memory
allocation
 The price of sharing is the contention for the shared
resource
 To reduce contention, the memory is usually split up
into modules, which can be accessed independently
and concurrently with other modules
 When more than one processor attempts to access to
the same module, only one processor can be granted
access, while other processors must await their turn
in a queue.
P. 36
Cheng-Fu Chou, CMLAB, CSIE, NTU
 The effect of such contention, or interference, is to
increase the average memory access time.
P. 37
Cheng-Fu Chou, CMLAB, CSIE, NTU
Memory modules
M1
●●●
Mj
●●●
Mm
Interconnection network
P1
●●●
Pi
●●●
Pn
Processors
P. 38
Cheng-Fu Chou, CMLAB, CSIE, NTU
Assumptions
 The time to complete a memory access is a constant
and all modules are synchronized
 Processors are assumed to be fast enough to generate
a new request as soon as their current request is
satisfied
 A processor cannot generate a new request when it is
waiting for the current request to be completed
P. 39
Cheng-Fu Chou, CMLAB, CSIE, NTU
 The operation of the system can be visualized as a
discrete-time queueing network
P. 40
Cheng-Fu Chou, CMLAB, CSIE, NTU
 The memory modules are the servers and the fixed
number, n, of processors constitute the jobs or
customers circulating in this closing queueing network.
 Let qi denotes the probability that a processor
generated request is directed at memory modules i, i=
1, 2, …, m.
 Consider a system with 2 memory and 2 processors
P. 41
Cheng-Fu Chou, CMLAB, CSIE, NTU
P. 42
Cheng-Fu Chou, CMLAB, CSIE, NTU
Memory Referencing Behavior
 In the renewal model, the successive intervals
between references to a given page were assumed to
be i.i.d. r.v..
 We consider a special case of the renewal model,
where the intervals are geometrically distributed.
This is known as the independent reference model
(IRM) of program behavior.
P. 43
Cheng-Fu Chou, CMLAB, CSIE, NTU
Independent Reference Model
 A program’s address space typically consists of
continuous pages represented by the indices 1,2,…,n.
 To study a program’s reference behavior, it can be
represented by the reference string w=x1, x2,…,xt.
 Successive references are assumed to form a
sequence of i.i.d. r.v. with
P(xt = i) = bi, 1  i  n.
 The interval between 2 successive references to page
i is geometrically distributed with parameter bi.
P. 44
Cheng-Fu Chou, CMLAB, CSIE, NTU
LRU model
 Assume that a fixed number, m (1m n), of page
frames have been allocated to the program.
 The state of the paging algorithm at time t, denoted
by q(t), is an ordered list of the m pages in main
memory.
 If the next page referenced (xt+1) is not in main
memory, then a page fault occurs. This will, in general,
require the replacement of an existing page from main
memory. We will assume that the rightmost page in
the ordered list q(t)
P. 45
Cheng-Fu Chou, CMLAB, CSIE, NTU
LRU model (cont.)
 If the next page referenced (xt+1) is in main memory,
no page fault (and replacement) occurs, but the q(t) is
updated to q(t+1)
 We can see that the sequence of states q(0), q(1),
…,q(t),… forms a DTMC with the state space
consisting of n!/(n-m)! Permutations over {1,2,…,n}
 Assume that the main memory is preloaded initially
with m pages.
P. 46
Cheng-Fu Chou, CMLAB, CSIE, NTU
LRU model Example
 Consider the LRU paging algorithm with n=3 and m=2.
 Let q(t) be ordered by the recency of usage, so that
q(t) = (i,j) implies that page indexed i was more
recently used than page j, and page j will be the
candidate for replacement.
P. 47
Cheng-Fu Chou, CMLAB, CSIE, NTU
P. 48
Cheng-Fu Chou, CMLAB, CSIE, NTU