On the Transient and Steady-State Analysis of a Special ... · On the Transient and Steady-State Analysis of a Special Single Server Queuing System with HOL Priority Scheduling Faouzi

196 On the Transient and Steady-State Analysis of a Special

ⓒGESTS-Oct.2005

On the Transient and Steady-State Analysis of a Special Single Server Queuing System with HOL Priority Scheduling

Faouzi Kamoun

Dubai University College, College of Information Technology, P.O. Box 14143 Dubai, United Arab Emirates

[email protected]

Abstract. In this paper, we consider a special discrete-time queuing system with two head-of-line (HOL) priority queues and a mix of correlated and uncor-related arrivals. The arrival process to the high priority queue is correlated and consists of a train of a fixed number of fixed-length packets, while the low pri-ority traffic consists of batch arrivals that are independent and identically dis-tributed from slot-to-slot. We derive an expression for the functional equation describing the transient evolution of this priority queuing system. This func-tional equation is then manipulated and transformed into a mathematical tracta-ble form. By applying the final-value theorem, the corresponding exact expres-sions for the steady-state marginal pgfs are derived. We also show how the tran-sient analysis provides insights into the derivation of the system’s busy period distribution. Finally, we illustrate our solution technique with some numerical examples, whereby we demonstrate the negative effect of correlation (in the high-priority queue) on the performance of the low-priority queue.

1 Introduction

ATM and next generation communications networks are being built around the moti-vation of having a single cost-effective packet-based network that is capable of sup-porting diverse classes of services, each with its own QoS requirement. To achieve this goal, various packet service disciplines that determine the order by which packets are served have been proposed. Among the simplest priority scheduling disciplines, the non-preemptive head-of-line (HOL) priority scheduling has been proposed to provide differentiated services to the high-priority traffic class. Under this scheme, whenever the server is idle, it always schedules the delay-sensitive traffic first (if pre-sent). In the literature, there have been various contributions towards the performance analysis of HOL-based priority systems. (see for example [1] , [2], [3], [4]).

This paper focuses on a special single-server discrete-time queuing system with two head-of-line (HOL) priority queues. The arrival process to the high priority queue is correlated and consists of a train of a fixed number of cells, while the low priority traffic consists of batch arrivals that are independent and identically distributed from slot-to-slot. Our work departs from the previous contributions, cited above in the lit-erature, in at least four aspects:

GESTS Int’l Trans. Computer Science and Engr., Vol.18, No.1 197

ⓒGESTS-Oct.2005

First, unlike previous studies that merely focused on the steady-state behavior of prior-ity systems, our analysis aims towards exploring the transient behavior of such sys-tems as well. As such, the present work can serve as a basis towards the transient analysis of other priority systems, under different arrival models and service-time distributions. Second, while the effect of correlation on the high-priority queue (with correlated arrivals) has been thoroughly investigated (see for example [5]), its impact on the behavior of the low-priority queue (fed by uncorrelated packet arrivals) has not been addressed. It is one of the objectives of this work to explore this important per-formance issue. Third, we show how the transient analysis leads to the characteriza-tion of the distribution of the busy period of the system, which is another distinctive contribution of this paper. Finally, our proposed approach which is purely based on probability generating functions is entirely analytical, does not involve any matrix concepts and provides a unified framework to extract transient as well as steady-state performance measures. The remaining of this paper is organized as follows: In sections 2 and 3, we describe the queuing model and present the functional equation for the joint pgf of the state vector. In section 4, the resulting functional equation is transformed into a new form that is mathematically tractable. In sections 5 and 6, we present the exact transient analysis. The corresponding steady-state results are derived in sections 7 and 8. In sections 9, we characterize the distribution of the busy period of the priority system. Numerical results that provide insights into the behavior of the queuing system are provided in section 10. Finally, a summary of the main findings of the paper and rec-ommendations for future research are provided in section 11.

2 Analytical Model Description

In this paper, we consider a discrete-time queuing system with infinite buffer capaci-ties and a single (FCFS) deterministic server. The time axis is divided into equal length slots and packet transmission is synchronized to occur at the slot boundaries. Here a slot is the time period required to transmit exactly one packet from the buffers. We consider two types of priority traffic, namely a high-priority and a low priority traffic (thereafter referred to as type-1 and type 2, respectively). The scheduling is based on a HOL time-priority scheme, whereby type-1 packets have absolute non-preventive priority over type-2 packets. Under this priority scheme, the server will always serve type-1 packets (if any) based on a FCFS basis. If there are no type-1 packets in the high-priority queue, then type-2 packets (if any) will be served; again based on a FCFS rule.

We model the arrival process to the high-priority queue by the correlated real-time traffic composed of a fixed-length packet-train arrival process. Correlated train arri-vals are encountered in various applications whereby customers are messages (e.g. Frames or jumbo packets) that consist of multiple fixed-length packets; see e.g. [6]. More precisely, the high-priority queue is fed with N input links, receiving fixed-length messages of m1 packets, each. These messages enter the system at a fixed rate of one packet per slot. In addition, traffic on different input links is assumed to be


ⓒGESTS-Oct.2005

independent and with the same statistical characteristics. On any input link, the prob-ability that the first (leading) packet of a message enters the buffer in any given slot is q if the first packet of the previous message on this link did not enter the buffer during the previous (m1-1) slots and it is zero otherwise. Further, let {cj ; j ≥1} be a series of independent and identical Bernoulli random variables with pgf 1 1( ) 1C z q qz= − + .

New message arrivals to the low-priority queue are assumed to be i.i.d. from slot-to-slot and characterized by the same pgf

2 2( ) kbB z E z⎡ ⎤= ⎣ ⎦ , independent of k. For sim-

plicity, it is assumed that that each type-2 message arrives in a batch of m2 packets; though our approach can be easily extended to handle variable-length messages. Fur-

ther, we define the load offered by class-j packets as jρ (j=1,2), while we denote by

1 2Tρ ρ ρ= + the total load to the system. For stability, we assume that 1Tρ < .

3 System Evolution and Preliminaries

The priority queuing model can be formulated as a discrete-time Markov chain. The

state of the system is defined by 11, 2, 1, 2, 1,( , , , ,.... )k k k k m ku u a a a −

where ,j ku are the system

contents of class j (j=1,2) queue at the end of the kth slot, while ,n ka1(1 1)n m≤ ≤ − is

the number of input links having sent the nth packet of a message to type-1 buffer in

slot k. The total system content 1, 2,( )k ku u+ will be denoted by ,T ku . Clearly [5]:

, 1 1, 1, 2n k n ka a n m+ −= ≤ ≤ (1)

1 1

1, 1 ,1 1

;kN I m

k j k n kj n

a c I a− −

+= =

= =∑ ∑ (2)

The evolution of the system contents is described by the following system equations

1

1, 1 1, 1, , 11

2, 2, 2 1 1,2, 1

2, 2 1 1,

[ ]

[ ] . if 0

. if 0

m

k k k n kn

k k k kk

k k k

u u u a

u u m b uu

u m b u

+

+ +=

+

+

+

+

= − +

⎧ − + =⎪= ⎨

+ >⎪⎩

∑ (3)

where [ ]x + is a random variable which takes the value 1 if x >0 and 0 otherwise.

Next, define the joint generating function of the priority system at the end of the kth

slot as follows:

1,1, 2 , 1, 1

1 11 2 1 2 1 2 1 1 2 1 1( , , ) ( , , , ,.. ) .. m kk k k au u a

k k m mQ z z x Q z z x x x E z z x x −

− −⎡ ⎤= = ⎣ ⎦

�

From the above and using (1-3), we can easily derive the following functional equa-tion relating the joint pgf of the system between two consecutive slots:


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[ ]2 2 1 2 1 2 11 1 2 2 1 1

1 2

1 2 2 1 2

1 2

( , , ( , ),.. ( , ))( , , ) ( ) ( . .

( ) (0, ,0,..0) ( 1) (0,0,..0)

Nm k mk

k k

z Q z z z x z xQ z z x B z C x z

z z

z z Q z z z Q

z z

+

⎧ ϒ ϒ= ⎨

⎩⎫− + −

+ ⎬⎭

�

(4)

where 11

1 1

.( , )

( . )i

i

x zz x

C x zϒ = and 1mx = for notational purposes.

From the above, it is clear that the presence of the 1( , )iz xϒ terms in the right-hand side

of (4) complicates the analysis. For instance, taking the limit as k → ∞ on both sides of (4) does not help in extracting the steady-state joint pgf of the system. In the sequel, we show how to handle the above functional equation to extract not only the steady-state but also the transient joint pgf of the queuing system.

4 Transforming the Functional Equation into a New Form

In this section, we show how to transform the functional equation (4) of the system into a new form that will lead itself to a solution. Without any loss of generality, we will assume zero initial conditions, whereby the system is initially empty, with all links being in an ‘idle’ state, i.e. 0 1 2( , , ) 1Q z z x =

� . Further, because of the Markovian prop-

erty of the system, the steady-state behavior is independent of the initial conditions.

With zero initial conditions and by expanding 1 1 2( , , )kQ z z x+

�in (4) for the first few

values of k, we can prove by simple induction the following major result which en-ables us to express 1 2( , , )kQ z z x

�in a more suitable form.

4.1 Theorem 1: Transient Joint PGF of the Priority System

Under zero initial conditions, the functional equation (4) describing the queuing model under consideration can be written as follows:

[ ] [ ]

[ ]∑

∑

=

−

=

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

k

jjk

jmN

xz

k

jjk

jmN

xz

kmN

xzk

Qz

zBjJ

z

zz

zQz

zBjJ

z

zz

z

zBkJxzzQ

1 1

2,

2

21

12

1

2,

2

21

1

2,21

)0,..,0,0()(

)()1(

)0,..0,,0()(

)()()(

)(),,(

2

1

2

1

2

1

�

��

(5)

where:


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ki

m

ixixz zzCkJ )()()( 1

11,,

1

1λ∑

=

= �� (6)

)(.)1)(1()(

)(.1

)( 1111

1

111

1,

1

zqmzm

zzxq

zC ii

m

j

ji

jj

xi λλ

λ

−−−

+

=

∑−

=

−

� (7)

and )( 1ziλ ’ s (i=1,2, ..m1) are the m1 distinct roots of the characteristic equation:

0)()1()( 1111

111 =−−− − mmm qzzqz λλ (8)

The proof of this theorem is by induction (in a similar way as in [7]) and will not be given here. Further, from (8), it is obvious that one of the m1 roots has the property that 1)1( =λ .This particular root is thereafter denoted by )( 11

zmλ .

5 Determination of the Transient Boundary Terms

To derive the transient boundary terms (0, 0,..0)kQ and 2(0, ,0..0)kQ z appearing in

(5), we proceed as follows:

First, let us define the following three transforms ( 1)w < :

1 2 1 2 2 20 0 0

( , , , ) ( , , ) ; ( ) (0,0,..0) ; ( , ) (0, ,0..0)k k kk k k

k k k

Q z z x w Q z z x w P w Q w R z w Q z w∞ ∞ ∞

= = =

= = =∑ ∑ ∑� � (9)

By substituting 1 2( , , )kQ z z x�

from (5) into 1 2( , , , )Q z z x w�

as defined above, and using

(6), we derive the following expression for the w-transform of the transient joint pgf of the system:

( )

( )

i

i

m

i

i

m

nm

ii

m

m

i

nixi

Nnnn m

m

nm

ii

m

m

i

nxi

Nnnn m

zzwBz

zzC

nnn

NwzB

z

wPzzwzRzz

zzwBz

zCz

nnn

NwxzzQ

�

�

�

�

�

��

1

2

1

121

2

2

1

121

1121

111,

... 212

2

2121

1121

11,1

... 12121

)()(

)()(.

!!...!.

!.)(

)()1(),2()(

)()(

)(..

!!...!.

!),,,(

=

=

=+++

=

=

=+++

−

⎥⎦

⎤⎢⎣

⎡ −+−+

−

=

∑

∑

λ

λ

λ

(10)

The above expression will not only be useful to derive the two transient boundary terms (0,0,..0)kQ and 2(0, ,0,..0)kQ z appearing in (5), but the steady-state results

related to our queuing model, as well.


ⓒGESTS-Oct.2005

5.1 Determination of the Transient Boundary Constants

The transient probabilities of an empty system (0,0,..0)kQ can be readily obtained

from (10) as follows:

First setting z1=z2=z in (10) yields the following expression for the w-transform of the

transient joint pgf of the total system contents ,T ku and the number of ‘active’ input

lines.

( )

( )

i

i

m

i

i

m

nm

ii

m

m

i

nixi

Nnnn m

m

nm

ii

m

m

i

nxi

Nnnn m

zzwBz

zzC

nnn

NwzBwPz

zzwBz

zCz

nnn

NwxzzQ

�

�

�

�

�

��

1

2

1

121 1

2

1

2

1

121 1

1

1,

... 21

1

1,

... 21

)()(

)()(.

!!...!.

!.)()()1(

)()(

)(.

!!...!.

!),,,(

=

=

=+++

=

=

=+++

−

−+

−

=

∑

∑

λ

λ

λ (11)

Next we determine ( )P w from the above expression by invoking the analytic property

of (11) inside the polydisk ( 1; 1)z w≤ < . For this purpose, let Nm zzH )()(

1λ= and

denote by ),(* wzYz = the unique root inside the unit circle of the equation:

0)().(. 2 =− zHzBwz m (12)

Then, following a similar approach as in [7], we can apply Rouchés theorem to show that

*1

1)(

zwP

−= (13)

Applying Lagrange’s theorem to (12-13), allows us to express the boundary function P(w) as follows:

[ ]0

21

1

1 )1(

)()(

!1)(

2

=−

−∞

= ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+= ∑

z

km

k

k

k

k

z

zHzB

dz

d

k

wwP (14)

which implies that:

21

1 20

( ) ( )1(0,0,0) ( 1)

! (1 )

kmk

k kz

B z H zdQ k

k dz z

−

−=

⎧ ⎫⎡ ⎤⎪ ⎪⎣ ⎦= ∀ ≥⎨ ⎬−⎪ ⎪⎩ ⎭

(15)

or equivalently ( using Leibniz’s rule for the kth derivative of a product) :


ⓒGESTS-Oct.2005

{ }21

0

( ) ( )1 ( )(0,0,0)

0!

kmik

k ii

d B z H zk iQ

zk i dz

−

=

⎡ ⎤⎣ ⎦−=

=∑ (16)

5.2 Determination of the Transient Boundary Functions

The boundary terms 2(0, ,0..0)kQ z appearing in (5) can be evaluated by exploiting the

analytical property of (10) inside the polydisk ( 1; 1)iz w≤ < (i = 1, 2), whereby

whenever the denominator of 1 2( , , , )Q z z x w

�vanishes, the numerator must also vanish

at the same values. We first derive the w-transform 2( , )R z w of this boundary term,

then take the inverse w-transform to obtain 2(0, ,0,..0)kQ z .

Again, Let Nm zzH )()( 11 1

λ= and for a given 2 1z < denote by ),( 2

*1 wzXz = the unique

root inside the unit circle 1 1z < of the equation:

0)().(. 1212 =− zHzBwz m (17)

From Rouchés theorem, it can be shown that

22

2222 ),(

)()1)(,(),(

zwzX

zwPzwzXwzR

−

−−= (18)

The transient boundary functions 2(0, ,0)kQ z can now be obtained by taking the in-

verse w-transform of (18), giving:

2 21

(0, ,0,..0) ( ) ( 1) ( ) (0,0,..0)k

k k jj

Q z k z j Q −=

= Γ + − Γ∑ (19)

Where:

2 ( )( ) 11 12 2( ) ( )0 ! 02 1 1

km i ki d H zB z zkk k i iik z i dz z

−Γ = −∑

= =

⎡ ⎤⎡ ⎤ ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

(20)

and (0,0,..0)kQ are as defined in (16).

From the above, it is clear that equations (5-7), (16) and (19-20) fully characterize the

transient joint pgf ( , , )1 2Q z z xk�

of the queuing system at the end of the kth slot.

We also note that since type-1 traffic is not influenced by the low-priority traffic, the high-priority queue could have been analyzed in isolation from the class-2 queue. A detailed analysis of the resulting single-class queue can be found in [7]. Our transient and steady-state results related to the higher- priority queue, which are omitted herein


ⓒGESTS-Oct.2005

due to lack of space, are also in agreement with the corresponding results derived in [7]. Thus, in the sequel, our analysis will mainly focus on the lower-priority queue.

6 Transient Analysis of Class-2 (Low-Priority) Queue

The marginal pgf 2, 2( )kP z of class-2 system contents of at the end of the kth slot is

readily obtained from (5):

[ ] [ ] ]0,..0,0()0,.0,,0([)()1(

)()(1

222

222,2

22

jk

k

jjk

jmkmk QzQzB

z

zzBzP −

=

− −−

+= ∑ (21)

where the boundary terms (0, 0, ..0)Qk and (0, , 0, ..0)2Q zk are as previously de-

fined in (16) and (19-20), respectively.

From the above marginal pgf, time-dependent performance measures can be derived. For instance, let 2,kN denote the mean queue length of class-2 queue at the end of the

kth slot. Then from (21) we can show that:

2

12, 2

2, 21 02

( )'(1) (0,1,0,..0) (0,0,..0)

kk

k j jz j

dP zN km B Q Q

z

−

= =

⎡ ⎤= = − −⎣ ⎦∑ (22)

7 Steady-State Joint PGF of the System

The steady-state joint pgf of the system is readily obtained by applying the final-value theorem to (10):

1 2 1 21( , , ) lim (1 ) ( , , , )

wQ z z x w Q z z x w−→

= −� �

(23)

or equivalently:

[ ]

( )

i

i

m

nm

ii

m

m

i

nixi

Nnnn m

m

zzBz

zzC

nnn

N

z

zB

QzzzQzzxzzQ

�

� �

�

1

2

1

121

2

1121

111,

... 212

2

2122121

)()(

)()(.

!!...!.

!)(.

)0,..0,0()1()0,..0,,0()(),,(

=

=

=+++−

−+−=

∑λ

λ (24)

The only unknowns in the above expression are the boundary terms

2(0, ,0,.0) and (0,0,..0)Q z Q . These can be readily determined from their transient

counterparts, as shown below.

The first boundary term Q(0,0,..0) is derived by applying the final value-theorem to (13):


ⓒGESTS-Oct.2005

1 1

1(0,0,..0) lim(1 ) ( ) lim

1 ( , )w w

wQ w P w

Y z w− −→ →

−= − =

− (25)

or equivalently:

12*

1

(0,0,0) 1 '(1)1 1 ( 1)

NqmdwQ m B

wdz m q= = − −

= + − (26)

This also implies that the total load of the system is

12

1

'(1)1 ( 1)T

Nqmm B

m qρ = +

+ −

Next, the second boundary term 2(0, ,0,..0)Q z appearing in (24) is derived by apply-

ing the final value-theorem to (18):

22

22

12 )(

)1)(()1(),(.)1(lim)0,.0,,0( 2

zzX

zzXwzRwzQ T

w −

−−=−=

−→ρ (27)

where, for given 2 1z < , )( 2zX is the unique root inside the unit circle

1 1z < of the

equation: 0)().( 1212 =− zHzBz m

Next, by substituting (27) into (24), we obtain the following equivalent expression for the steady-state joint pgf of the system:

( )

i

i

m

nm

i

im

m

i

nixi

Nnnn m

mT

zzBz

zzC

nnn

N

zzX

zXzzBzxzzQ

�

� �

�

1

2

1

121

2

1121

111,

... 21

22

212221

)()(

)()(

.!!...!.

!.

)(

)()()1)(1(),,(

=

=

=+++−

−

−−−=

∑λ

λ

ρ

(28)

8 Steady-State Analysis of Class-2 (Low- Priority) Queue

The marginal pgf 2 2( )P z of class-2 system contents is readily obtained by setting

z1=xi=1 (i=1,..m1-1) in (28); giving:

( ) ( ))(1

)(

)(

)(1)1)(1()(

2

2

2

2

22

2222 m

m

TzB

zB

zzX

zXzzP

−−

−−−= ρ (29)

Alternatively, by setting z1=xi=1 (i=1,..m1-1) in (24), we can express 2 2( )P z in terms

of the two boundary terms, as follows:


ⓒGESTS-Oct.2005

)(1

)()0,..0,0()0,..0,,0()1()(

2

2

2

2

2

2222 m

m

zB

zB

z

QzQzzP

−⎥⎦

⎤⎢⎣

⎡ −−= (30)

Applying the normalization condition, P2(1)=1 to (30) , yields

11 2 1 2

1

and . '(1)1 ( 1) T

Nqmm B

m qρ ρ ρ ρ= = − =

+ −

Next let 2N denote the mean buffer occupancy of the low-priority queue in steady-

state. By differentiating (30) twice with respect to z2 and setting z2=1 in the resulting expression, we get after few intermediate steps:

22 2 2

2 2 21

1 ''(1) ( 1)''(1) ''(1). .

1 2(1 ) 2(1 ) 2(1 ) 2(1 )T T T T

m B mH HN

ρρ ρ

ρ ρ ρ ρ ρ

− −= + − + +

− − − − − (31)

where:

2

2 11 1 1

( 1)''(1) ( 1)

NNH m

N N

ρρ ρ

−− ⎡ ⎤= + − ⎢ ⎥⎣ ⎦

9 Busy Period Analysis of the Priority Queuing System

In order to characterize the probability distribution of the busy period of the priority system, we will use the technique described in [2]. Specifically, let:

( ) ( 0,1,2,....)j pr b j slots j⎡ ⎤Ω = = =⎣ ⎦��

denote the probability that the system is busy for j slots.

Assuming that the last time the priority-system was empty at the end of the (k-j)th slot, we can express the transient probabilities of an empty system as:

1

(0,0,..0) ( 1) (0,0,..0) ( 0)k

k k jj

Q j Q k−=

= Ω − >∑ � (32)

Next, define the following w-transform: 0

( ) ( ) j

j

B w j w∞

=

= Ω∑� � . By substituting for

Qk(0,0,..0) as in (32) into P(w) as given in (9) we get:

( ) 1( )

( )

P wB w

wP w

−=� (33)

Since in general we define the busy period of a queuing system as the time between two consecutive idle periods, then the busy period must consist of at least one slot (i.e. initiated by at least one arrival, which occurs with probability1 (0)(1 )NB q− − ). Un-


ⓒGESTS-Oct.2005

der this convention, let the random variable b denote the length of an arbitrary busy period in number of slots and let ( )B w be the corresponding pgf. Clearly:

1. ( ) ( 1)

1 (0)(1 )( ) [ ]

0 ( 0)

Nj j

B qj probability b j slots

j

⎧Ω ≥⎪

− −Ω = = = ⎨⎪ =⎩

� (34)

It follows that:

* (0)(1 )( )

1 (0)(1 )

N

N

z wB qB w

w B q

− −=

⎡ ⎤− −⎣ ⎦ (35)

where z* is as defined in (12). From the above pgf, we can use Lagrange’s theorem to derive the expression for the corresponding probability mass function ( )jΩ :

2

1( ) . ( )1 1

( ) . . ( 0)01 (0)(1 ) ( 1)!

jmj

N j

d B z H zj j

zB q j dz

+⎡ ⎤⎣ ⎦Ω = >

=− − +

(36)

10 Numerical Examples and Discussions of the Results

In this section, we illustrate our approach through some numerical examples, and probe further into the interplay between the correlation of type-1 traffic and the performance of class-2 queue. All the numerical results presented were obtained using Maple™ [8] computational software. We have assumed that the number of message arrivals to the low-priority (type-2) queue follows a Geometric Batch arrival process,

characterized by the pgf 22 2

2

( ) 1 (1 (1 ))B z zm

ρ= + − . In figure 1, we plot the transient mean

queue length ,i kN (i=1,2) of type-i buffer, as well as the transient mean of the total

system content ,T kN .

In particular, we note that while the exponential rise behavior in the transient mean-time curve of type-1 queue, is typical in many other queuing systems, the transient mean-time curve of type-2 queue exhibits a sharper ‘linear’ growth behavior. This can be explained by the earlier observation that all of the 2 4m = packets of a type-2 mes-

sage enter their buffer during the same slot, whereas type-1 messages enter their buffer at the fixed rate of one packet/slot.


ⓒGESTS-Oct.2005

Fig. 1. Transient means of buffer occupancies ( 1 2 2 10.8; 0.1; 4; 2; 8m m Nρ ρ= = = = = )

Next to investigate the effect of type-1 correlation on the behavior of type-2 queue, we plot in figure 2, the steady-state mean buffer content of type-2 queue versus the total load 1 2Tρ ρ ρ= + for different values of type-1 correlation indicator (m1). Here,

we assumed 2 5, 5,m N= = and fixed 2 0.2ρ = .

Fig. 2. Steady-state mean buffer occupancy of the low-priority queue

As may be seen from figure 2, the correlation factor m1 (associated with the high-priority buffer) has a significant influence on the mean buffer occupancy of type-2 queue. For the same traffic loads and traffic parameters, 2N increases rapidly as m1

increases. The above leads us to conclude that the correlation of type-1 traffic has a direct negative impact not only on the high-priority queue it is feeding, but also on the low-priority queue. Therefore prior studies which ignored this correlation effect should be revisited as they might lead to inaccurate buffer dimensioning, admission and congestion control policies.

0

20

40

60

80

100

120

140

160

180

200

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Total load

Ste

ady-

stat

e m

ean

bu

ffer

o

ccu

pan

cy

m1=1

m1= 4m1= 8

m1= 16

m1= 32

m1= 642N

0

20

40

60

80

100

120

140

160

180

200

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Total load

Ste

ady-

stat

e m

ean

bu

ffer

o

ccu

pan

cy

m1=1

m1= 4m1= 8

m1= 16

m1= 32

m1= 64

0

20

40

60

80

100

120

140

160

180

200

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Total load

Ste

ady-

stat

e m

ean

bu

ffer

o

ccu

pan

cy

m1=1

m1= 4m1= 8

m1= 16

m1= 32

m1= 642N

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25 30 35

Time (number of slots)

Tra

nsi

ent

mea

n o

f q

ueu

e le

ng

th

1,kN

2,kN

,T kN

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25 30 35

Time (number of slots)

Tra

nsi

ent

mea

n o

f q

ueu

e le

ng

th

1,kN

2,kN

,T kN


ⓒGESTS-Oct.2005

11 Conclusions and Suggestions for Further Research

In this paper, we have proposed an alternate solution technique towards the transient and steady-state analysis of a special non-preemptive HOL priority queuing system. Our work revealed that in a HOL-based priority system, the correlation of packet arrivals to the high-priority queues affects not only these queues, but the lower-priority queues as well. This work can be further explored in many directions. For example, variants to our queuing model, such as different arrival processes, finite buffers and multi-server cases can be investigated. These are left for future research.

References

[1] J. Walraevens, and H. Bruneel, Performance Analysis of a Single-Server ATM Queue with Priority Scheduling, Computers and Operations Research, Vol. 30, No. 12, 2003, pp. 1807-1830.

[2] M. Mehmet Ali and X, Song, A Performance Analysis of a Discrete-Time Priority Queuing System with Correlated Arrivals, Performance Evaluation, No. 57, 2004, pp. 307-339.

[3] K. Laevens, and H. Bruneel, Discrete-Time Multi-server Queues with Priorities, Per-formance Evaluation, No. 33, 1998, pp. 249-275.

[4] R. Jafari and K. Sohraby, Performance Analysis of a Priority Based ATM Multiplexer with Correlated Arrivals, IEEE Infocom 99, 1999, pp. 1036-1043.

[5] Y. Xiong and H. Bruneel, Performance of Statistical multiplexers with Finite Number of Inputs and Train Arrivals, in proceedings of INFOCOM, 1992, pp. 2036-2044.

[6] H. Bruneel, Packet Delay and Queue Length of Statistical Multiplexers with Low-Speed Access Lines, Computer Networks, Vol. 25, 1993, pp. 1267-1277.

[7] F. Kamoun, Performance Analysis of a Discrete-Time Queuing System with a Corre-lated Train Arrival Process , Performance Evaluation, In Press.

[8] Maple is a registered trademark of Maplesoft (http://www.maplesoft.com/)

Biography

▲ Faouzi Kamoun P.O. Box 14143. Dubai. UAE. Holds a Ph.D degree in Electrical and Computer Engineering from Concordia University, and an MBA degree in Man-agement from McGill University, Canada. He joined the College of IT at Dubai University College in September 2002, as an Assistant Professor. Prior to that, he has been with Nortel Networks (Canada) since 1995, where prior to his departure in 2002, he was a Senior Technical Advisor in the Hi-CAP Optical Networks Division. His research interests are in the modeling and performance analysis of communications net-works. Tel: +971-4-2242472 E-mail: [email protected]

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On the Transient and Steady-State Analysis of a Special ... · On the Transient and Steady-State Analysis of a Special Single Server Queuing System with HOL Priority Scheduling Faouzi