Generalized Processor Sharing with Long-Range ... - Semantic Scholar

Generalized Processor Sharing with Long-Range Dependent Traffic Input Xiang Yu, Li-Jin Thng, Yuming Jiang Department of Electrical & Computer Engineering

Abstract In this paper, we develop an upper bound on the individual session queue length of long range dependent (LRD) traffic under the generalized processor sharing (GPS) scheduling discipline. This work is based on the analysis of the arrival process of LRD traffic, which is demonstrated to be Weibull Bounded Burstiness (W.B.B.) process in our paper. Decomposing GPS system into isolated queues and servers, we are able to obtain the bound on the individual session queue length from their arrival processes. Our work proves that, in different scenarios, different parameters of the upper bound of an individual session queue, such as the index, the asymptotic constant and the decay rate can be affected by other flows existing in the GPS system. Under certain conditions, by carefully choosing the weight parameters, an individual session with LRD traffic input can be well isolated from other flows in a GPS system.

1. Introduction Future high speed digital networks are expected to support a variety of applications such as voice, video and data with different traffic characteristics and quality of service (QoS) requirements. Hence there is a need for traffic control mechanisms to regulate network resources in order to achieve differentiated QoS. In particular, the study of the scheduling disciplines to be employed in network switches becomes one important issue. An ideal scheduling discipline should satisfy two requirements: (1) Provide isolation between sessions. This guarantees that the scheduling discipline is able to protect individual flow against misbehavior of other flows. (2) Realize statistical multiplexing gain. This suggests the flows can utilize each other’s excess service rate. Perhaps the most widely studied non-FCFS scheduling discipline is the Generalized Processor Sharing (GPS) discipline, which is first studied by Parekh and Gallanger [11], [12]. GPS has two attractive characteristics: (1) each backlogged session is guaranteed a minimum service rate. This guarantees that the misbehavior of other flows has limited effects on an individul session and provides the foundation of isolation between sessions. Achieving isolation further enables GPS to guarantee differentiated QoS for individual session. (2) it is work-conserving, and the excess service rate is redistributed among the backlogged flows in proportion to their weight parameters. The second characteristic enables GPS to obtain statistical multiplexing gain between input flows. Because of these two characteristics, GPS is deemed as an ideal scheduling that can satisfy the two requirements mentioned

above and attracts a lot of researchers’ interest. When GPS is extended to packet switched networks, it is usually referred to as Weighted Fair Queuing (WFQ) or Packet-based GPS (PGPS) [7], [18]. However, achieving differentiated QoS in GPS scheduling still remains as a challenging task because the asymptotic behavior of GPS with different bursty traffic input can be different, especially when the input traffic is different from the traditional sources whose tail distribution of queue length has an exponential form. Most of the work on GPS carried out in the past was based on the assumption that the input traffic streams of a network satisfies some burstiness constraints, either deterministic ones, such as Leaky-Bucket bounds [15], or stochastic ones, such as the Exponential Bounded Burstiness (E.B.B.) process [18], [20]. The traditional stochastic traffic whose tail distribution of queue length has an exponential form is demonstrated to have an E.B.B. arrival process [19]. Based on such constraints on the arrival process, an upper bound has been developed for individual session queue length in GPS system for traditional input sources [18], [20]. In contrast to traditional traffic, a long range dependent (LRD) traffic with burstiness extending over various time scales has different tail distribution of queue length as well as arrival process. The asymptotic behavior of GPS with LRD traffic input can be different from those with traditional traffic of E.B.B. input process [7]. Recent research of Borst [4], [5], [16] has shown that under certain conditions a long range dependent traffic can be well isolated from other input flows in GPS system, and in some other scenarios, it may inherit bursty properties from other flows. The issue of studying isolation problem for individual long range dependent flow becomes the focus of implementing GPS with LRD traffic input. Until we are able to guarantee isolation for an admitted long range dependent flow in GPS can we study the multiplexing gain between admitted flows. On the other hand, LRD traffic is becoming an important class of traffic in modern day networks. Research has discovered that long range dependence property exists in Ethernet traffic [9], ATM traffic, compressed video traffic [8], [3] and so on. Hence, it is important to study the behavior of GPS scheduling discipline with LRD traffic input. The objective of this paper is to study the asymptotic behavior of GPS with LRD traffic input, or, to be more precise, to study the tail distribution of queue length of individual LRD traffic in GPS system. We aim at getting an upper bound on the tail distribution, and by analyzing such an upper bound, we are able to study the isolation issue as mentioned above. Long range dependent traffic is usually modeled from its tail distribution of queue length, but in this paper we study its behavior in GPS

system from its arrival process. It is demonstrated that the tail distribution of queue length obeys some type of Weibull distribution [3], [2], and the queue length distribution is also referred to as Weibull Bounded (W.B.) process in our paper. Based on such assumption, we propose a burstiness constraint on its arrival process, which is demonstrated to be a Weibull Bounded Burstiness (W.B.B.) process in our paper. Assuming the input traffic of a GPS system with long range dependence in nature having a W.B.B. arrival process, we decompose the GPS system into isolated queues and servers by using the methods developed in [20] and obtain an upper bound for individual session queue length. With such an upper bound, we are able to study the isolation problem and multiplexing gains in GPS scheduling discipline with LRD input traffic. The rest of the paper is organized as follows. In section 2, we introduce certain notations in GPS scheduling discipline which we will use later in the paper. In section 3 we introduce Weibull Bounded Burstiness (W.B.B.) process and Weibull Bounded (W.B.) traffic model, its relationship with LRD sources and some of its properties. We examine the sample path behavior of sessions for a single GPS server with LRD traffic input and develop an upper bound on their queue lengths using the decomposed GPS system in section 4. In section 5, we extend the results obtained in section 4 from GPS server to PGPS server. Finally we conclude the paper in Section 6.

2. Preliminaries Generalized Processor Sharing (GPS) is a scheduling discipline defined under the assumption that sources are described by fluid models. Consider a GPS server with rate γ serving N sessions, each session i is assigned a weight parameter, which is a fixed real-valued positive number φ i , where {φ 1 , φ 2 ,..., φ N } is called a GPS assignment. The N sessions share the server in the following way [12]: 1. It is work conserving, i.e. the server keeps serving the queues if there are packets backlogged in the GPS system. 2. The excess service rate is redistributed among the backlogged sessions in proportion to their weight parameters. 3. If session i is backlogged in the system during the whole interval [τ , t ] , i.e., in the interval [τ , t ] there is always traffic queued for that session at all times, let S k (τ , t ) be the amount of session k traffic served in a time interval [τ , t ] , then Si (τ , t ) φ i ≥ , j = 1,2,..., N . (1) S j (τ , t ) φ j From (1), it is clear that when session i is backlogged, it is guaranteed a backlog clearing rate (or equivalently, a guaranteed service rate) of at least φi γ. gi = (2) N φj

∑

j =1

For simplicity, we will assume γ = 1 throughout the paper. To study the relationship between backlogs from a set of sessions, we need to give an ordering to them according to their

arrival rates and weight parameters, We then give the definition of feasible ordering that will be used in our paper later.

∑

Definition 1. Given

N

i =1

ρ i < 1 , there exists an ordering

among the sessions such that, after relabeling:

ρi
w} < Ce −ηw

υ

(4)

The definition of W.B.B. arrival process is given as follows. Definition 3. Let A(t) be a traffic arrival rate. A(t) has a Weibull Bounded Burstiness (W.B.B) with parameter ρ , C,

η , υ (or ( ρ , C , η , υ )-W.B.B.) if it satisfies

1

t

P{ A(u)du > ρ (t − s) + w} < Ce −ηw

υ

(5)

s

for all w ≥ 0 and all 0 ≤ s < t . Remark: In the above two definitions, we call C the asymptotic constant, η the decay rate and υ the index parameter. In Definition 2, ρ is called the long term upper rate of the arrival rate. Long-range dependent source usually has the property that its backlog or queue length has some kind of W.B. tail distribution ([3], [6], [14]) with the index parameter υ = 2(1 − H ) , where H is Hurst parameter, which is used to characterize the degree of long range dependence. However, most of the studies of LRD traffic didn’t give the characteristics or properties of the arrival process of LRD traffic. In Theorem 1 that will be introduced later, we demonstrate the relationship between a traffic source with such a W.B. tail distribution and its arrival process.

Next, we develop several theorems that are useful in studying the asymptotic behavior of GPS system with LRD sources. Theorem 1. Consider a system that transmits at rate ρ , and assume it is work-conserving. Suppose that it is fed with a single stream of traffic rate A(t), and let W(t) be the amount of work stored in the system at time t. We have: (i) If W(t) is W.B, then A(t) is W.B.B with upper average rate ρ. (ii) If A(t) is W.B.B, with upper average arrival rate ρ − ε for some ε > 0 then W(t) is W.B. Proof : For (i), by the assumption that W(t) is W.B., there exist some positive C, η and υ ( 0 < υ ≤ 1 ) with P{W (t ) > w} < Ce −ηw

υ

for all w ≥ 0 and all t ≥ 0 . Let 0 ≤ s < t . Since the system can transmit no more than ρ (t − s) during the time from s to t, then we have

1

t

{W (t ) > w} ⊇ { A(u)du > ρ (t − s) + w} .

1

( ρ 2 , C2 , η 2 , υ 2 )-W.B.B. The two processes can be either dependent

or

independent.

Then

A1 (t ) + A2 (t )

where α =

η1η 2 η1 + η 2

is and

υ = min(υ 1 , υ 2 ) .

t

P{ A(u)du > ρ (t − s) + w} s

≤ P{W (t ) > w} < Ce −ηw

Lemma1. Let A1 (t ) be ( ρ 1 , C1 , η1 , υ 1 )-W.B.B. and A2 (t ) be

( ρ 1 + ρ 2 , C1 + C2 , α , υ )-W.B.B.

s

Hence

Remark: Long range dependent traffic is characterized with the W.B. tail distribution of queue length [8], [2]. In Theorem 1, we demonstrated that long range dependent traffic generally has an W.B.B. arrival process. When we study the asymptotic behavior of GPS scheduling with LRD traffic input, or the tail distribution of queue length in the buffer of GPS server, we can study from its arrival process instead of studying directly from the tail probability distribution of individual queue length. This result will be used later to develop an upper bound on queue length from the queue lengths of decomposed GPS system with LRD traffic input. Notice that in (7), the asymptotic constant will not be affected by the index parameter υ , which means in the tail distribution of backlog of a flow with a W.B.B. arrival process, the asymptotic constant is immune from the index, or the Hurst parameter, i.e., the LRD degree of the input flow. This property is useful in Theorem 2 when we study the isolation issue of GPS with LRD input flows.

Proof: Set 0 ≤ s < t and w ≥ 0 , and let 0 < p < 1 be some constant. Then, we have

υ

1 1 1 1 1 1 t

{ ( A1 (u) + A2 (u))du > ( ρ 1 + ρ 2 )(t − s) + w}

A(u) is a ( ρ , C , η, υ )-W.B.B. process.

s

t

⊂ { A1 (u)du > ρ 1 (t − s) + pw} 

To prove (ii) we begin with the condition that A( t ) is W.B.B., i.e.

1

t

P{ A( u) du > ( ρ − ε )( t − s) + w} < Ce −ηw

υ

is true for all w ≥ 0 and all 0 ≤ s < t . Since we are only interested in W (t ) > w , the backlog W(t) is larger than zero and thus can be expressed as follows: s0 < t

1

t

{ A2 (u)du > ρ 2 (t − s) + (1 − p) w}

(6)

s

W ( t ) = max{

s

t

s

and thus

t

P{ ( A1 (u) + A2 (u))du > ( ρ 1 + ρ 2 )(t − s) + w} s

s0

s t

Hence there exists some s0 < t such that W (t ) =

1

t

s0

t

≤ P{ A1 (u)du > ρ 1 (t − s) + pw}

A(u )du − ρ ( t − s0 )} .

≤ P{ A2 (u)du > ρ 2 (t − s) + (1 − p) w} s

A( u) du − ρ ( t − s0 )

< C1e

−η 1 pwυ1

+ C2 e −η 2 (1− p ) w

υ2

and if we assume W (t ) > w then we have

1

t

s0

Choose p such that η 1 p = (1 + p)η 2 . Then p =

A( u) du > ρ ( t − s0 ) + w .

1

And we have

Setting α =

t

P{W (t ) > w} = P{ A(u)du > ρ (t − s0 ) + w}

1

s0

t

= P{ A(u)du > ( ρ − ε )(t − s0 ) + ε (t − s0 ) + w} s0

υ

< Ce −η[ w + ε ( t − s0 )] < Ce −2

−1

ηwυ −2 −1 ηε υ ( t − s0 )υ

≤ Ce −2

−1

ηw υ

1

η2 . η1 + η 2

η1η 2 and υ = min(υ 1 , υ 2 ) we get η1 + η 2

t

P{ ( A1 (u) + A2 (u))du > ( ρ 1 + ρ 2 )(t − s) + w} s

υ1

< C1e −αw + C2 e −αw ≤ (C1 + C2 )e

υ2

(8)

−αwυ

e

Lemma 2. Let A1 (t ) be ( ρ 1 , C1 , η1 , υ 1 )-W.B.B., A2 (t ) be (7)

Proof of (ii) in Theorem 2.2 from (7).

( ρ 2 , C2 , η 2 , υ 2 )-W.B.B. and assume they are independent.

Then, A1 (t ) + A2 (t ) is ( ρ 1 + ρ 2 , C12 , η − ψ , υ )-W.B.B. where

η = min(η 1 , η 2 ) and υ = min(υ 1 , υ 2 ) , and C12 and ψ depend on C1 , C2 . Proof: Set 0 ≤ s < t and w ≥ 0 . Assume further that A1 ( s, t ) is the workload arrived in the time interval ( s, t ) , and it has a distribution density, to be denoted by a1s,t , such that

1

P{ A1 ( s, t ) > w} =

∞

w

a1s,t (u)du = 1 −

1

w

0

a1s,t (u)du

1 1

q = 0 P{ A2 ( s, t )

=

∞

q =0

1

a1s,t (q ) × P{ A2 ( s, t ) > ( ρ 1 + ρ 2 )(t − s) + w − q}dq

υ

υ2

υ2

+ (C2 + C2η 2 e −η 2 [ ρ1 ( t − s)] )e −η 2 w υ2

ρ 1 ( t − s)

q =0

+ C1e −ηw

υ2

+ C1e −η1w

υ1

υ1

(15) Lemma 2 follows from (15) if we choose C21 and ψ such that υ2

υ

(16)

Notice that there are some approximations in the above proof using Inequality 1 and Inequality 2 that are demonstrated in details in Appendix of this paper.

a1s,t (q ) × P{ A2 ( s, t ) > ρ 2 (t − s) + w}dq

< C2 e −η 2 w

υ2

C2 (2 + η 2 e −η 2 [ ρ1 ( t − s)] ) + C1 ≤ C21eψw

a1s,t (q ) × P{ A2 ( s, t ) > ( ρ 1 + ρ 2 )(t − s) + w − q}dq

< P{ A1 ( s, t ) < ρ 1 (t − s)} × C2 e −η 2 w

υ2

4. Sample Path Behavior of GPS Server with Long Range Dependent Traffic Input

υ2

(10) ρ 1 ( t − s) + w

1

q = ρ1 (t − s)

1

q = ρ 1 ( t − s) ∞

q

In this section we study the sample path behavior of the sessions served by a GPS server with LRD traffic input. We assume that each session has its own infinite queue. From the results of section 3, we suppose that LRD traffic for each session, whose tail probability of queue length or backlog is characterized with a W.B. distribution, has a W.B.B. arrival process Ai with long term upper rate ρ i such that

a1s,t ( q ) × P{ A2 ( s, t ) > ( ρ 1 + ρ 2 )(t − s) + w − q}dq

ρ 1 ( t − s) + w

≤ (−

1

∞

< C2 e −η 2 w

= P{ A1 ( s, t ) < ρ 1 (t − s)} × P{ A2 ( s, t ) > ρ 2 (t − s) + w}


( ρ 1 + ρ 2 )(t − s) + w}

> ( ρ 1 + ρ 2 )(t − s) + w − q}

a1s,t (q ) × P{ A2 ( s, t ) > ( ρ 1 + ρ 2 )(t − s) + w − q}dq

ρ1 ( t − s)

≤

a1s,t (q ) × P{ A2 ( s, t ) > ( ρ 1 + ρ 2 )(t − s) + w − q}dq

< (C2 + C2η 2 e −η 2 [ ρ1 ( t − s)] )e −η 2 w

< C1e −η1w

Splitting the above integral in (9) into 3 parts, and using integration by parts, we get

1

q = ρ1 ( t − s)

≤ a1s,t (q )dq = P{ A1 ( s, t ) > ρ 1 (t − s) + w}

P{ A1 ( s, t ) ∈[q , q + dq )} ×

∞

ρ1 ( t − s) + w

q = ρ1 ( t − s) + w

Then P{ A1 ( s, t ) + A2 ( s, t ) > ( ρ 1 + ρ 2 )(t − s) + w} =

1

then the above inequality can be written to:

υ2

a1s,t ( q )C2 e −η 2 ( ρ1 ( t − s) + w− q ) dq υ2

a1s,t ( u) du) × C2 e −η 2 ( ρ1 ( t − s) + w− q ) |qρ=1 (ρt −(st)−+sw) +

ρ 1 ( t − s) + w

q = ρ1 (t − s)

1

1

υ2

C2η 2υ 2 q υ 2 −1e −η 2 {[ ρ1 ( t − s) + w]

≤ C2 e −η 2 w

υ2

1

∞

a1s,t ( q )dq − C2

ρ1 ( t − s)

υ2

C2η 2 e −η 2 [ ρ1 ( t − s) + w] ≤ C2 e −η 2 w

υ2

+

1

w

q =0

υ2

C2η 2 e −η 2 [ ρ1 ( t − s)] e −η 2 w = C2 e −η 2 w

υ2

υ2

1 1

w

q

∞

υ2

∑

a1s,t ( u)dudq

∞

q + ρ 1 ( t − s)

υ2

w

ρ i < γ = 1 . By abuse of notation, let Ai also denote a

sample path (or a realization) of a random arrival process Ai ,

a1s,t (u)dudq

so Ai (τ , t ) is the amount of traffic from session i during the time interval [τ , t ] on this sample path. Similarly, we will use Qi to denote the corresponding sample path of the corresponding random process of queue length or backlog Qi .

υ1

υ2

e −η1 ( ρ1 ( t − s) + q ) e η 2 ( ρ1 ( t − s) + q ) dq

e −[η1 ( ρ1 ( t − s) + q )

υ1

−η 2 ( ρ 1 ( t − s ) + q ) υ 2 ]

q=0

ρ 1 ( t − s) ≤ q ≤ ρ 1 ( t − s) + w

N

i =1

a1s,t (q ) dq +

1

ρ 1 ( t − s)

eη 2 ( ρ 1 ( t − s) + q )

∞

q=0

+

C2η 2 e −η 2 [ ρ1 ( t − s)] e −η 2 w

If η1 ≥ q υ 2 −υ 1 η2

υ2

1

− qυ2 }

(11)

(12)

dq

Imagine that we divide a GPS server of rate γ = 1 into a set of N (fictitious) servers with rate γ 1 , γ 2 ,..., γ N , so that, instead of having a GPS system sharing a server, we have a decomposed system consisting of N separate queues, each of which has a dedicated server. We use δ i and γ i to denote the queue length and service rate for each session i in the decomposed system (see Figure 1). We then aim at developing bounds on the actual session i backlog Qi of the real GPS system in terms of the fictitious session i backlog δ i of the imaginary decomposed system.

δ (t )

Q1 ( t )

Q2 ( t )

δ 2 (t )

γ =1

γ1 γ

2

server queue can be applied. In our case where the arrival process characterization is given in the form of a W.B.B. process introduced in Section 3, The proof of Theorem 1 (7) provides an upper bound on δ i (t ) which is paraphrased below: Lemma 4. For any q ≥ 0 P{δ i (t ) ≥ q} ≤ Ce −2

δ N (t )

QN (t )

N

First, we choose the set of γ i in our paper according to [20], which satisfies the following requirements : 1. 2. 3.

∑

i =0

γ i ≤1

∑

N j =i

(1 −

φj

∑γ

j),

( ρ i , Ci , η i , υ i )-W.B.B. processes sharing a GPS server with

j =1

∑

N ε i =1 i

and γ i = ρ i + ε i , ≤ 1−

∑

N i =1

ρ i . If the

ϕ iυ j η j η i' , j −1

≥q

υ i' , j −1 −υ j

, 1 ≤ j ≤ i − 1 (19)

ρ j ( t − s) ≤q ≤ ρ j ( t − s) + w

then at any time t, for any q > 0 , o υ io

P{Qi (t ) > q} < Cio e −ηi q

(20)

where

η io =

1 min(η i , ϕ i υ j η j ) j =1,...,i −1 , 2

(21) (22)

i −1

φi j =i

ρ i < 1 and that 1,2,..., N is a

υ io = min(υ i , υ j ) j =1,...,i −1

Lemma 3. For any t,

∑

N

i =1

following condition holds

An important upper bound on individual queue length or backlog is given as [20]:

N

∑

i = 1,2,..., N , where ε i > 0 and

1≤ i ≤ N

Such a set of γ i is called a feasible ordering as defined in Definition 1. The first requirement guarantees that such a feasible ordering of the sessions always exists. The second requirement makes sure that ε i = γ i − ρ i > 0 so that each decomposed queuing system is stable. In the rest of the paper, we assume that 1,2,…,N is a feasible ordering of N sessions with respect to {r1 , r2 ,..., rN } .

Qi (t ) ≤ δ i (t ) +

Theorem 2. Suppose that Ai ( i = 1,2,..., N ) are N independent

feasible ordering with respect to φ i

i −1

φi

(18)

system in terms of the fictitious queue length δ i (t ) in the decomposed GPS system with the upper bound of the fictitious queue length (18) of the decomposed GPS system, we are able to obtain the upper bound of the queue length in GPS system.

assignment φ i . Assume

ρi < γ i

γi ≤

ηw υ

Combining the upper bound (17) of queue length Qi (t ) in GPS γ

Figure 1 Decompose GPS system

N

−1

φj

∑δ

j (t )

(17)

j =1

Remark: Lemma 3 gives an upper bound on queue length Qi (t ) of individual session in a GPS system in terms of the queue length δ i (t ) of the fictitious queue in the decomposed GPS system. To see the proof of Lemma 3, please refer to [20]. Because Lemma 3 is independent of whatever stochastic model used to characterize each session’s input traffic, we will use them to establish the upper bound on the tail distribution of the queue length or backlog for each session with LRD traffic input (or traffic with W.B.B. arrival process) in this paper. From Lemma 3, we see that in order to bound the tail distribution of Qi (t ) we only need to bound δ i (t ) for any time t. As δ i (t ) is the queue length or backlog of session i when it is served by a server with rate γ i , known results for the single

i

Cio =

∑C ∏ j

j =1

Proof: Let ϕ i =

j −1 l =1

( 2 + η i ,l e

φi

∑

N j =i

φj

υ i ,l

−η i ,l [ ρ l ( t − s )]

)

(23)

. From (17) in Lemma 4, we have

P{Qi (t ) ≥ q} ≤ P{δ i (t ) + ϕ i

i −1

∑δ

j (t )

≥ q}

(24)

j =1

As the backlog δ i only depends on its arrival process Ai up to time t and all the other arrival processes A1 , A2 ,..., AN are independent, backlogs δ 1 , δ 2 ,..., δ N are also independent for any fixed t. Because the arrival processes for {δ 1 , δ 2 ,..., δ N } are all independent W.B.B. process, under the condition (30) and extending the proof of Lemma 2 (15) from two independent processes to i independent processes, we have

P{ Ai (t ) + ϕ i

i −1

∑

A j (t ) > ( ρ i + ϕ i

j =1

< [Ci

∏

i −1 j =1

i −1

∑ [C ∏ j

j =1

≤ Ci ,i −1e

j −1

( 2 + η i ,l e

−η i ,i −1q

∑

study the isolation issue, we may ignore the change made on the asymptotic constant Ci by other admitted flows.

ρ j )t + q}

j =1

( 2 + η i , j − 1e

l =1

i −1

υi , j −1

−η i , j −1 [ ρ j ( t − s )]

υi ,l

−η i ,l [ ρ l ( t − s)]

υi

)] × e −η i q +

)] × e

−η jϕ i υ j q

To extend Theorem 2 from the point of the definition of LRD traffic, or from a source given in terms of queue length distribution, we give the following Theorem 3:

υj

Theorem 3. Supposing that N independent LRD flows with ( Ci , η i , υ i )-W.B. tail distribution of queue length Wi

υi ,i −1

(25) where for 1 ≤ k ≤ i − 1 , Ci , k = Ci k

∏

∑ [C ∏ j

j =1

k j =1

(2 + η i , j −1e

j −1 l =1

( 2 + η i ,l e

υ i −1

−η i , j −1 [ ρ j ( t − s )]

∑

than the system link capacity or equally )+

γ i = ρi + ε i ,

∑

)]

N

ε i =1 i ϕ iυ j η j

η i ,k = min(η i , ϕ i υ j η j ) j =1,...,k

(27)

υ i ,k = min(υ i , υ j ) j =1,..., k

(28)

η i , j −1

N

i =1

≤ 1− ≥q

i = 1,2,..., N ,

∑

N i =1

υ i , j −1 −υ j

where

ρ j ( t − s) ≤q ≤ ρ j ( t − s) + w

then at any time t, for any q > 0 ,

< Ci ,i −1e

υi ,i −1

(31)

i −1

∑δ

j (t )

> q}

j =1

−2 −1 η i ,i −1q

and

, 1 ≤ j ≤ i − 1 (30)

o υ io

P{Qi (t ) > q} ≤ P{δ i (t ) + ϕ i

εi > 0

ρ i . If the following condition holds

P{Qi (t ) > q} < Cio e −ηi q

Using Lemma 4, it is easily to prove that

ρ i < 1 and that

is a feasible ordering with respect to φ i and

1,2,..., N (26)

υi ,l

−η i ,l [ ρ l ( t − s)]

( i = 1,2,..., N ) sharing a GPS server with assignment φ i . Assume the sum of the average arrival rate of all flows is less

=

(29)

where η io = min(η i , ϕ i υ j η j ) j =1,...,i −1 ,

(32)

υ io = min(υ i , υ j ) j =1,...,i −1

(33)

o

o υi Cio e −ηi q

Theorem 2 gives an upper bound on an actual session i backlog Qi (t ) of the real GPS system in terms of the traffic characteristics of other admitted sources in GPS system. When the tail distribution of backlogs of other sessions in a GPS system is lighter than or equal to that of session i, i.e., υ j ≥ υ i , j ≠ i , υ io approaches υ i . It means that the other admitted sources with lighter tails will not affect the queuing behavior of session i on the index parameter υ i , or equally the Hurst parameter H ( Hi = 1 − υ i / 2 ) and thus will not affect the long range dependence property of session i. However, those lighter tailed sources do affect the queuing behavior of session i on the asymptotic constant Ci and the decay rate η i . From equation (32) it is clear that if the weight parameters are carefully chosen, it is possible to keep the decay rate η i of session i unchanged too. This means the session i traffic can be deemed as isolated from other flows in GPS system from other admitted flows in bursty characteristics. The asymptotic constant is however a little complicated and is affected by traffic characteristics of other flows such as the average arrival rates ρ j and the decay rates η j , and GPS weights φ j as well, but it will not be affected by the index parameter υ j , or the LRD property of other admitted flows. However, the asymptotic constant doesn’t affect the bursty characteristics or long range dependent property on the backlog or queue length of session i input traffic and will not introduce induced burstiness from other sessions as discussed in [5]. So, when we

i

Cio =

∑C ∏ j

j =1

j −1 l =1

(2 + η i , j e

υ i ,l

−η i , j [ ρ l ( t − s )]

)

(34)

See the Appendix for proof.

5. Results Extended to a PGPS server One evident limitation of the GPS discipline is that it assumes that the input traffic is fluid model and it serves multiple sessions simultaneously or bit by bit. Such service discipline is different from practical implementation. An attempt to overcome this limitation is the Packet based GPS (PGPS), which approximates GPS in the more practical case, when the server can only handle one packet at a time. In packet switched networks, a server will serve a packet only when its last bit arrives. So, a PGPS server also considers a packet as arrived after its last bit has been received, and it uses a simulation of GPS to determine the order at which arrived packets receive service. Whenever a PGPS server becomes free, it selects for service the arrived packet that would be the first to complete service in the GPS simulation, if no additional packet were to arrive. Unlike the GPS server, a PGPS server does not start serving a packet if it is only partially received. In other words, the PGPS is a Feed and Forward part, while GPS is Cut Through. To cope

with this difficulty, we separate the PGPS server into two imaginary parts – a regulator and a PGPS core (see the discussion in Chapter 4 of [10]). Here, the function of the regulator is to perform the Store and Forward part. It collects the recerved data into packets, and passes only complete packets to PGPS core. The output of this regulator, which is the input to the PGPS core, is a series of impulses, whose heights represents the sizes of the packets (See Figure 2).

Regulator PGPS Server

( H j = 1 − υ j / 2 ) are smaller. By carefully choosing the GPS

Figure 2 PGPS Server Let Ai be the session i input traffic to the PGPS server, which is also the input to the regulator part, and Ai is the output traffic from the regulator, which is the input traffic to the PGPS core. We have Ai ( s, t ) = li ( s) + Ai ( s, t ) − li (t ) (35) where li (τ ) is the amount of data belonging to a session i packet that is partially received in the regulator by time τ . Let L be the maximum length of all arrival packet. Since 0 ≤ l (τ ) ≤ L , we conclude that

Ai ( s, t ) ≤ Ai ( s, t ) + L

(36)

Considering a PGPS server with W.B.B. inputs, we can employ (36) to (5) and get P{ Ai ( s, t ) > ( ρ i − ε )(t − s) + q} ≤ P{ Ai ( s, t ) > ( ρ i − ε )(t − s) + q − L} < Ci e −ηi ( q − L ) υi

υi

≤ Ci eηi L e −η i q

6. Conclusion In this paper, we analyzed the asymptotic behavior of the Generalized Processor Sharing scheduling discipline with long range dependent traffic input and developed an upper bound on the queue length or backlog of each session. From the derived upper bound of the queue length for an individual session, we conclude that the long range dependent source of an individual session i in GPS system will generally keep its LRD property despite the presence of other admitted sources with lighter tails (i.e., the index υ i ≤ υ j , j ≠ i ), or whose Hurst parameters H j

PGPS Core

R

So, when Theorem 2 is extended to PGPS server, the PGPS server can also be treated as a GPS server with similar W.B.B. arrival processes only with a little modification on the asymptotic constant from that of the W.B.B. arrival processes to GPS server according to (37).

weight parameters, it is possible to keep the decay rate of the individual LRD source unchanged. This result proves the conclusion in recent research on GPS with long tailed traffic input [4] and [5] using an alternative method. What’s more, we also extended the result into Packet based GPS (PGPS) system, which approximates GPS in practical use. which suggests that the results developed in GPS can be easily extended for practical use. The conclusion in our paper suggests that under certain conditions, GPS-based scheduling algorithms are able to provide an effective mechanism for isolating individual LRD flow from other admitted bursty flows. With such isolation achieved in GPS server with LRD input flows, GPS-based scheduling algorithms are able to extracting multiplexing gain, while protecting individual LRD source. In particular, for a call admission control (CAC) algorithm of heavy-tailed LRD sources that uses effective bandwidth estimation, the estimation could be based on its own tail distribution when all the other admitted sources are lighter tailed and the weight parameter is carefully chosen.

(37)

7. Reference

υi

[1]

The last step of (37) assumes that q > L , the assumption from the Inequality 3 in the appendix of this paper, which assumes that the time period [ s, t ] is enough for L workload to be received. This assumption means that the result for the arrival process or the upper bound extended to the queues in PGPS sever is useful only when the queue length or backlog is large enough to exceed one maximum packet length L. In the other word, the extended results are useful under large buffer size scenarios. Such an assumption is quite reasonable because this is most of the cases. In such a case, when the input to the PGPS server is W.B.B., the input to the PGPS core is also W.B.B. with the same decay rate η i and index υ i , and with the υi

asymptotic constant changed from Ci to Ci eη i L .

[2]

[3]

[4]

[5]

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Appendix: This appendix gives proofs of inequalities used in this paper. For all y1 , y2 ,..., y N > 0 and 0 ≤ x < 1 , we have Inequality 1: y1x + y2x +...+ y Nx ≥ ( y1 + y2 +...+ y N ) x Inequality 2: y1x + y2x +...+ y Nx ≤ N ( y1 + y2 +...+ y N ) x Inequality 3: y1 x − y2 x ≤ ( y1 − y2 ) x , for all y1 > y2 . Proof:

Suppose f ( x ) = y x , 0 ≤ x < 1 and y > 0 . Then,

f ' ( x ) = ( y x )' = (e x ln y )' = e x ln y × ln y (39) Since y >1 >0 ln y = { , y 0 f ' ( x) = { y y2 , y1 − y2 > 0 , from Inequality 1 we have:

( y1 − y2 ) x + y2 x ≥ [( y1 − y2 ) + y2 ] x = y1 x from which we get ( y1 − y2 ) x ≥ y1 x − y2 x , which gave the proof of Inequality 3.