Mar 6, 2009 - aCentre for Quantifiable Quality of Service in Communication Systems (Q2S) ... results set up the basis for a network calculus for gSBB traffic.
Fundamental Calculus on Generalized Stochastically Bounded Bursty Traffic for Communication Networks ?
Yuming Jianga , Qinghe Yinb , Yong Liuc, Shengming Jiangd,e a Centre
for Quantifiable Quality of Service in Communication Systems (Q2S) and Department of Telematics Norwegian University of Science and Technology (NTNU), Norway b Australian Institute of Health and Welfare, Australia c Department of Electrical and Computer Engineering, National University of Singapore, Singapore d School of Electronic & Information Engineering, South China University of Technology, China e Faculty of Advanced Technology, University of Glamogran, UK.
Abstract Since many applications and networks do not require or provide deterministic service guarantees, stochastic service guarantee analysis is becoming increasingly important and has attracted a lot of research attention in recent years. For this, several stochastic versions of deterministic traffic models have been proposed in the literature. Unlike previous stochastic models that are based on the traffic amount property of an input process, we present another stochastic model, generalized Stochastically Bounded Burstiness (gSBB), which is based on the virtual backlog property of the input process. We show the advantages of this approach. We study the superposition of gSBB traffic, and set up the input-output relation. Under various service disciplines, we characterize the output process for each source and investigate probabilistic upper bound on delay. Finally, we introduce a stochastic ordering monotonicity property of gSBB. With this property, we show that many well-known traffic models can be readily represented using the proposed gSBB model. These results set up the basis for a network calculus for gSBB traffic. Key words: Network calculus, generalized Stochastically Bounded Burstiness (gSBB), stochastic service guarantees. ? The material in this paper was partially presented at IEEE LCN 2002 [38] and ITC19 2005 [19].
Preprint submitted to Elsevier Science
6 March 2009
1
Introduction
The increasing demand on transmitting multimedia and other real time traffic over the Internet has motivated the study of providing service guarantees in the Internet. For this study, many traffic models have been proposed, which can be put into two categories: deterministic traffic models and stochastic traffic models. The (σ, ρ) or token bucket constrained traffic model [9] is perhaps the most widely studied deterministic traffic model due to the seminar work of Cruz [9][10]. The (σ, ρ) model has been generalized to the concept of envelope or arrival curve for deterministic service guarantee analysis [6][24]. Because deterministic service guarantee can result in (possibly very) low network resource utilization and because many applications and networks do not require or provide deterministic service guarantees, stochastic service guarantee has emerged as a promising candidate for supporting quality of service (QoS) in the Internet. For studying stochastic service guarantee, one crucial issue is traffic modeling. In the past years, researchers have proposed several stochastic traffic models extending the deterministic (σ, ρ) model to the stochastic case, which include exponentially bounded burstiness (EBB) [37], stochastically bounded burstiness (SBB) [35] that generalizes EBB, and statistical envelope [4][3][8]. The idea behind most of these models is to use some bounding functions to statistically upper-bound the traffic amount of a flow and use these functions to model the flow. This is consistent with the idea of the (σ, ρ) model where a function deterministically upper-bounding the traffic amount, which is referred to as the traffic amount property in this paper, is used to model the flow. While this way of extending the (σ, ρ) model to the stochastic case is intuitively simple, recent study has revealed that to apply these resulting stochastic traffic models to stochastic service guarantee analysis needs special care [38]. In [35][2], some restriction is enforced on the proposed stochastic traffic model itself to ensure that the resulting one can be used for deriving stochastic service guarantees. In [25], the difficulty of using the directly extended stochastic traffic model to analyze stochastic service guarantee is highlighted, and an approach is proposed in [25] to address the difficulty. In this paper, we explore another property of the (σ, ρ) model, which, referred to as the virtual backlog property in this paper, is that the queue length of a virtual single server queue (SSQ) fed with the same flow is upper-bounded. It can be proved that the virtual backlog property is equivalent to the traffic amount property of (σ, ρ) model. In the paper, we further show that there is similar duality principle under stochastic settings. Based on these, we define a new stochastic traffic model, called generalized Stochastically Bounded Burstiness (gSBB). Its idea is to use some functions to statistically upper-bound the queue length distribution in a virtual SSQ with input from a flow, and use 2
these functions to model the flow. We study the superposition of gSBB traffic, and set up the input-output relation [38]. In addition, the present paper further develops the basic results of [38]. Particularly, we also consider a work-conserving system shared by a number of gSBB sources. Under different service disciplines, we characterize the output process for each source and investigate the bounding probability for delay. Finally, we introduce a stochastic ordering monotonicity property of gSBB. With this property, we show that many well-known traffic models can be readily represented using the proposed gSBB model. The rest of the paper is organized as follows. In the next section, the network model and some background on traffic modeling are introduced. In addition, we define gSBB and study its duality principle in this section. In Section 3, with the proposed stochastic traffic model, some basic results for stochastic service guarantee analysis are derived. In Section 4, we study the superposition of gSBB traffic, and set up the input-output relation. In Section 5, we analyze the characteristic of the output process for each source and investigate the bounding probability for delay under various service disciplines. In Section 6, we prove the monotonicity property of gSBB and show that various existing traffic models belong to gSBB. Finally in Section 7, concluding remarks are given.
2
Network Model and Background
We consider a discrete time system. We assume that servers are work-conserving, by which we mean a server becomes idle only when there is no customer waiting for service in the server. We further assume that the size of each network buffer is infinite. We also assume that the system starts at time t = 0 and all the network queues are empty at the beginning time t = 0. Before introduction on various traffic models, we introduce some notations to denote some frequently used expressions in short forms. We use A(s, t) for the traffic amount arriving in the time interval (s, t], i.e. t X
A(s, t) =
a(i) ,
i=s+1
where we adopt the convention A(t, t) = 0. For ease of expression, we use A(t) to represent A(0, t). In addition, we use a(t) to represent traffic arriving in (t − 1, t] or a(t) = A(t − 1, t) = A(t) − A(t − 1). 3
ˆ ρ) which is defined as The next notation we introduce is A(t; ˆ ρ) = sup {A(s, t) − (t − s)ρ} A(t; s≤t
where ρ is a constant, often representing a rate. ˆ ρ) is indeed If ρ is the capacity of a constant server with input A(t), then A(t; the queue length at time t, and we have h
i+
ˆ + 1; ρ) = A(t; ˆ ρ) + a(t + 1) − ρ A(t
.
Here, we use [·]+ to express the maximum of 0 and a given number. We shall also use [·]1 to denote the minimum of 1 and the given number, i.e., [x]1 =
x
if x ≤ 1, if x > 1.
1
References [9], [10], [32] and [33] set up the mathematical basis for deterministic service guarantee analysis, with which the deterministic bounds on delay and buffering requirements can be obtained. In them, the input process A(t) of studied traffic has the following property: A(s, t) ≤ (t − s)ρ + σ ,
(1)
for all s ≤ t, where ρ and σ are constants. We may say that the input process A(t) has a “Deterministically Bounded Burstiness” (DBB). This model is also referred to as the (σ, ρ) model which defines that the amount of arrival traffic in any time interval is upper bounded by a function. This traffic amount property has been utilized to find its stochastic counterpart in literatures. The first direct generation is exponentially bounded burstiness (EBB) [37] which is defined as follows. A flow is said to be EBB iff for all s ≤ t, and all x > 0, P {A(s, t) − (t − s)ρ > x} ≤ ae−bx
(2)
where ρ, a and b are some constants. The concept of EBB has been extended to stochastically bounded burstiness (SBB) in [35] to generalize ae−bx to a function f . The function f belongs to a function class F which contains all the functions defined on [0, ∞) with the following properties: (i) f ∈ F is non-negative and non-increasing; R (ii) for f ∈ F, letting f (1)(x) = x∞ f (u)du, then f (1) ∈ F. The definition of SBB is as follows [35]: 4
Definition 1 A flow is said to be stochastically bounded bursty with upper rate ρ and bounding function f iff there exists f ∈ F and for all 0 ≤ s ≤ t and all x > 0, there holds for the arrival process A(t) P {A(s, t) > (t − s)ρ + x} ≤ f (x) .
(3)
Another related concept is effective envelope [4][3] or statistical envelope [8]. A traffic stream is said to have a statistical envelope g(t) iff the following inequality is satisfied for all τ ≥ 0 and t ≥ 0: P {A(t, t + τ ) ≤ g (τ )} ≥ 1 −
(4)
where ε is a constant that may be interpreted as the targeted violation probability. In fact, letting = f (x) and g (τ ) = τ ρ + f −1 () in the statistical envelope model will result in the SBB model, where f −1 (·) denotes the inverse function of f (·). As we can see from (2), (3) and (4), all these models tend to directly characterize the statistical behavior of traffic amount. However, the analytical results of SBB are not applicable if the bounding function in (3) is required only to be non-negative non-decreasing [25]. Such functions, including f (x) = x−α , indeed form a (possibly much) larger bounding function class than F . In order to relax the restriction on the bounding function and allow to use the larger bounding function class, we introduce in the following the concept of generalized Stochastically Bounded Burstiness (gSBB). For gSBB, we only require that the bounding function is non-negative and non-decreasing. It should be noticed that this requirement on the bounding function is much looser than that on the function class F in Definition 1. As a result, gSBB can be used to characterize wider range of traffic types than SBB. We shall see that SBB traffic is also gSBB. However, for gSBB traffic to be SBB, the bounding function f (x) needs to be in the function class F . It is shown in [35] that under SBB, if f (x) is the bounding function of an input process, a bounding function, which is the sum of f (x) and a constant R multiple of the function f (1) (x) = x∞ f (u)du, is needed for the output process. This is the reason why [35] required the bounding function f (x) to be in the function class F . Because of this “divergent” property of bounding functions for SBB traffic, after the traffic passes through a number of servers, its bounding function under the SBB model would become very large, as will be shown later by an example in Section 4. For gSBB traffic, we prove that the output process can use the same bounding function as the input process. In fact, we show that the output traffic of a constant rate server can be less bursty in the sense that the maximal cumulative burstiness is smaller. 5
3
Definition and Duality Principle
In this section, we first introduce the duality principle of (σ, ρ) model, based on which, we define the new traffic model called generalized Stochastically Bounded Burstiness (gSBB). Then, we show a similar duality principle for gSBB under certain conditions.
3.1
Duality Principle of (σ, ρ) Model
ˆ ρ) as follows: For flow A, we have defined function A(t; ˆ ρ) = sup{A(s, t) − (t − s)ρ}. A(t;
(5)
s≤t
ˆ ρ) is as follows. Let us consider a virtual single A useful interpretation of A(t; server queue (SSQ) system fed with the flow, which has infinite buffer space and is initially empty. Suppose the virtual SSQ is a constant rate server with rate ρ. Then, it is clear that the unfinished work or backlog in the SSQ system ˆ ρ). We call this virtual backlog property of A. Formally, for at time t is A(t; ˆ ρ), the following result is easily verified: A(t) and A(t; Theorem 1 For any x ≥ 0, A(s, t) − (t − s)ρ ≤ x for all 0 ≤ s ≤ t, if and ˆ ρ) ≤ x for all t ≥ 0. only if A(t; Theorem 1 reveals that the traffic amount property of (σ, ρ) model, represented by the first part, is equivalent to its virtual backlog property represented by the second part. Or, if the former holds, so does the latter and vice versus. We hence call Theorem 1 the duality principle of (σ, ρ) model.
3.2
Definition and Duality Principle of gSBB
The definition of gSBB is based on the virtual backlog property. In particular, we define gSBB as follows: Definition 2 A flow is said to have gSBB with upper rate ρ and bounding function f , denoted by A(t) hf, ρi, iff for all t ≥ 0 and all x ≥ 0, there holds ˆ ρ) > x} ≤ f (x), P {A(t; (6) where f is a non-increasing function and f (x) ≥ 0 for all x ≥ 0. 6
We now present a similar duality principle of gSBB as the (σ, ρ) model, whose proof can be found from the appendix. For this, we adopt the concept of stochastic ordering between two random variables. Specifically, random variable (respectively vector) X is said to be stochastically smaller than random variable (respectively vector) Y , denoted by X ≤st Y , iff for all x, P {X > x} ≤ P {Y > x} [31] [34]. Theorem 2 Let X be some non-negative random variable. ˆ ρ) ≤st X for all t ≥ 0, then A(s, t)−(t−s)ρ ≤st X for all 0 ≤ s ≤ t. (i) If A(t; (ii) If A(s, t) − (t − s)ρ ≤st X for all 0 ≤ s ≤ t and particularly {A(0, t) − ρ · ˆ ρ) ≤st X for all t ≥ 0. t, . . . , A(t − 1, t) − ρ} ≤st {X, . . . , X}, then A(t; In Theorem 2, by letting f (x) = P {X > x}, the first part, i.e. A(s, t) − (t − s)ρ ≤st X, implies the definition of SBB, and the second part, i.e. sups≤t {A(s, t) − (t − s)ρ} ≤st X, implies the definition of gSBB. Comparing Theorem 1 with Theorem 2, we can see that the former is more general than the latter in the sense that less restriction or assumption is needed for establishing the duality principle of arrival curve. In addition, while Theorem 1 shows that the traffic amount property of (σ, ρ) model is equivalent to its virtual backlog property, the duality principle of gSBB holds only in the context of stochastic ordering and with some additional requirements on A(t). The requirements for the second part of Theorem 2 to hold seem to be very restrictive. Because of this, we establish the following more general duality principle of gSBB. Its first part follows trivially since for all 0 ≤ s ≤ t, we always have A(t − s, t) − (t − s)ρ ≤ sups≤t {A(s, t) − (t − s)ρ}; for the second part, the proof follows the same approach as for Theorem 3. ii) in [35] and can also be found from [38]. Theorem 3 (i). If A(t) (ρ, f) for some f ∈ F, then A(t) is SBB with the same upper rate and bounding function. (ii). If A(t) is SBB with upper rate ρ and bounding function f ∈ F, then for any > 0, we have A(t) (ρ + , [g]1 ), (7) where g (x) = f (x) +
1
Z
∞
f (u)du.
(8)
x
Remark: The above theorem tells that if a traffic source can be modeled by SBB, it also can be modeled by gSBB (but may be with a larger bounding function). In addition, gSBB may be used to model traffic that does not have a bounding function f ∈ F. For example, gSBB can be used to model traffic 7
whose tail distribution has a power function [13] [16]. Particularly, Jelenkovi`c in [15] proved that for a server with capacity ρ, if its traffic input A(t) has distribution function F (x) = 1 − x−αl(x) where l(x) is a slow variation function, such as log x, and ρ is larger than E[a(t)], then its queue length, which ˆ ρ), is bounded by is indeed A(t; ˆ ρ) > x} ≤ K1 x−α , P {A(t;
(9)
where K1 > 0 is a constant. Based on Definition 2, it is clear that A(t) is gSBB with upper rate ρ and bounding function K1 x−α .
4
Superposition and Input-Output Relation
In this section we first consider the sum of two gSBB processes. We show that the sum is also a gSBB process with the sum of the upper rates of the summands as its upper rate, but its bounding function is not a simple sum of the two. Then, we consider the input-output relation for a constant rate server. We prove that if the input process is gSBB then the output process is gSBB with the same upper rate and same bounding function as the input process. The proofs can be found from [38]. Theorem 4 (Superposition) Assume A1 (t) (ρ1 , f1) and A2 (t) (ρ2, f2 ). Then (10) A1(t) + A2(t) (ρ1 + ρ2 , gp ), (i) where, we have gp (x) = [f1 (px) + f2 ((1 − p)x)]1 ,
(11)
and p can take any value in (0,1). (ii) If A1 and A2 are independent, we have gp (x) = 1 − F1 ? F2 (x) where Fi (x) = 1 − fi (x) for i = 1, 2, and ? denotes the Stieltjes convolution: R F1 ? F2 (x) ≡ 0x F1 (x − y) dF2 (y). we can get that A1(t) + Example 1. Let A1(t), A2(t) h(ρ, e−x ). In general, i −px −(1−p)x A2(t) (2ρ, gp ) where gp (x) = e +e , for any 0 < p < 1. When 1 p = 1/2, we get the minimum g1/2(x) = [2e−x/2]1. If A1 (t) = A2(t), then the bounding function for the aggregate process A1(t) + A2(t) is g(x) = e−x/2, and g(x) = 12 g1/2(x) for x ≥ 2 ln 2. If A1(x) and A2 (x) are independent, we 8
calculate the convolution of 1 − e−x with itself: Z
x
(1 − e−(x−y) )e−y dy =
0
Z
x
(e−y − e−x )dy
0
=1 − (1 + x)e−x . Hence when A1(t) and A2 (t) are independent, the bounding function is h(x) = (1 + x)e−x , which decays faster than e−αx for any α < 1. This example implies that we can get better results with the independent assumption. Next we consider input-output relation. We shall show that if an input process is gSBB with upper rate ρ and bounding function f , after passing through a work-conserving system with constant service rate C > ρ, the output process is also gSBB with the same upper rate ρ and the same bounding function f . In fact we have the following stronger result, whose proof can be found from [38]. Theorem 5 (Input-Output Relation) Consider a work-conserving system with constant service rate C. Let A(in) (t) and A(out)(t) be the input and output processes of the system respectively. Assume that A(in) (t) (ρ, f) for some ρ > 0 and non-negative non-increasing function f . Then for any t > 0 with probability 1 we have Aˆ(out) (t; ρ) ≤ Aˆ(in)(t; ρ). (12) With Theorem 5, we get immediately the following result: Corollary 1 With the same assumption as Theorem 5, if for some ρ > 0 and some non-negative non-increasing function f we have A(in) (t) (ρ, f), then with probability 1, A(out)(t) (ρ, f). Remark. Theorem 5 tells that the outgoing stream is less bursty than the incoming stream in the sense that the up-to-date maximal cumulative burstiness is smaller. Theorem 5 is consistent with similar findings in the literature under different traffic models (e.g. [28], [14], and Lemma 1.4.2 (iii) of [6]). Nevertheless, we want to highlight that Theorem 5 can hold only when there is no cross traffic. We will see in the next section that if cross traffic exists, the bounding function of the out-going traffic may become much larger. Example 2. Assume that we have an incoming stream A0(t) (ρ, e−x ), which is passing through a series of work-conserving servers with service rate C = ρ + 1. Use Ai(t) to denote the outgoing stream from the i-th server. By the Corollary of Theorem 5, we have Ai (t) (ρ, e−x ) for all i ≥ 0. If we consider A0(t) as SBB traffic and apply Theorem 3(i) of [35], then we can only get that the bounding function of A1(t) is given by −x
f1 (x) = e
+
Z
∞ x
9
e−u du = 2e−x .
In general, the bounding function of Ai(t) under SBB model is fi (x) = 2i e−x . If we consider i = 10, for x = 10 ln 2 ≈ 6.931, we could not obtain any useful information from the above result. In fact, we can only say that P {A10(s, t) ≥ (t − s)ρ + 10 ln 2} ≤ 210 e−10 ln 2 = 1.
However, by the Corollary of Theorem 5, we have P {A10(s, t) ≥ ρ · (t − s) + 10 ln 2} n o ˆ10(t; ρ) ≥ 10 ln 2 ≤P A ≤e−10 ln 2 = 2−10 < 0.001.
From the above example we see that besides that the extent of gSBB traffic is larger than that of SBB traffic, the conservative feature of gSBB for its bounding functions is a significant advantage comparing with the divergent feature for bounding functions with the SBB model. Particularly, the resulting bounding function from SBB input-output relation analysis will become larger and larger as the number of hops increases according to Theorem 3(i) of [35]. However, from gSBB input-output relation analysis presented here, the resulting bounding function remains unchanged regardless the number of hops in the absence of cross traffic. One implication of the gSBB input-output relation is that for a traffic stream passing through a network, if its initial SBB characteristic is known, we may first convert it to gSBB based on Theorem 3.(ii) and then apply the gSBB input-output relation to analyze its network performance. As illustrated in the above example, although the converted gSBB bounding function for the stream at the first server may be larger (than its initial SBB bounding function), the resulting bounding function after the last hop from gSBB input-output relation analysis could be much smaller than what would be obtained from SBB input-output relation analysis. Particularly, in the above example, the gSBB bounding function at the first server is 2e−x . From the gSBB inputoutput relation, we can conclude that after passing through the ten servers, the output traffic still has a gSBB (as well as SBB) bounding function 2e−x . However, if the SBB input-output relation had been used, the output traffic would have a SBB bounding function 210 e−x that is 29 times larger. 10
5
Stochastic Service Guarantees under Different Service Disciplines
In this section we consider a work-conserving server shared by multiple input processes. We shall use the results of previous sections to characterize output traffic and delay for each source under different service disciplines. Since we consider discrete model, we do not distinguish the order of arrivals which arrive at the same time. We assume that upon service, ties are broken arbitrarily among arrivals that have arrived at the same time. When workload arriving at time t and also being served at time t, we consider the delay as 0. The error between the delay defined in this way and the realistic delay is within one time unit. Use D(t) to denote the maximal delay for elements arriving at time t. D(t) = n means that the last bit of A(t) is transmitted at time t + n.
5.1
General Results
In this part we do not assume any particular service discipline. We consider the behavior of the outgoing stream for each source. Theorem 6 Suppose that we have N sources sharing a work-conserving system with service rate C. Assume that Aj (t) (ρj , fj ), where Aj (t) is the input P (out) (t) be the output process of process from source j, and N j=1 ρj < C. Let Aj (out) source j. Then for any ε > 0, we have Aj (t) (ρj , [g,j ]1 ) where g,j (x) = gj (x) +
1 ε
Z
∞ x
gj (u) du
and gj (x) = fj (pj x) +
N X
fi (pi x)
(13)
i=1
for any pi ∈ (0, 1), i = 1, . . . , N , satisfying pj +
PN
i=1
pi = 1.
PROOF. We use A(t) to denote the aggregate input process, i.e., A(t) = i=1 Ai (t). By Theorem 4, we have A(t) (ρ, g) where
PN
ρ=
N X
ρi ;
(14)
i=1
g(x) = [f1 (q1 x) + f2 (q2x) + · · · + fN (qN x)]1 11
(15)
for any qi ∈ (0, 1) satisfying q1 + q2 + · · · + qN = 1. Use Q(t) to denote the queue length at time t and Qi (t) to denote the portion of the queue which ˆ ρ). Notice that for any s, belongs to source i. Clearly, Qi (t) ≤ Q(t) ≤ A(t; (out)
Aj
(s, t) ≤ Qj (s) + Aj (s, t),
due to the fact that the output in the time interval (s, t] cannot exceed the sum of the input in this interval and the amount of workload from the same source stored in the queue previously. Then (out)
Aj
(s, t) − (t − s)ρj ≤ Qj (s) + Aj (s, t) − (t − s)ρj ˆj (t; ρj ) ≤ Q(s) + A ˆ ρ) + A ˆj (t; ρj ) ≤ A(s;
By a similar discussion as in the proof of Theorem 4, we can get that (out)
P {Aj
(s, t) − (t − s)ρj ≥ x} ≤ gj (x),
(16)
where gj (x) = g(px) + fj ((1 − p)x) = fj ((1 − p)x) +
N X
fi (pqi x)
i=1
with 0 < p < 1. We can get the form of (13) by letting pqj = 1 − p, i.e., 1 p = 1+q , and pi = pqi , i = 1, 2, · · · , N. According to the definition of SBB j (out)
and Inequality (16), Aj is SBB with upper rate ρj and bounding function gi . Then according to the relationship between SBB and gSBB from Theorem (out) 3, we have Aj (t) (ρj , [g,j ]1 ). Remark. The approach of the proof above is to derive the SBB bounding function first for the output process and then its gSBB bounding function is obtained by the relationship between SBB and gSBB. One may ask why the gSBB bounding function cannot be obtained directly for the output. If we try to consider Aˆ(out) (t; ρ) first, then by the definition of gSBB, there exists a special s∗ ≤ t such that (out) (out) Aˆj (t; ρj ) = Aj (s∗ , t) − ρj · (t − s∗ ) (out)
By definition of gSBB, s∗ is a variable that makes Aj (s, t) − ρj · (t − s) (out) is random, this s∗ is reaches its maximum value among all s ≤ t. Since Aj also random in nature. Following the same approach in the proof above, given s∗ is known, we can have (out) ˆ ∗; ρ) + Aˆj (t; ρj ). Aˆj (t; ρj ) ≤ A(s
12
Given s∗ , a bounding function for the right hand side of the above inequality can be obtained. In other words, it is possible to obtain a bound on (out) (out) P {Aˆj (t; ρj ) > x|s∗ = s}. However, what we want to have is P {Aˆj (t; ρj ) > x} that is indeed: (out)
P {Aˆj
(t; ρj ) > x} =
t X
(out)
P {Aˆj
(t; ρj ) > x|s∗ = s} · P {s∗ = s},
s=0
since s∗ is a random variable. Unfortunately, P {s∗ = s} is generally unknown. This the reason why the gSBB bounding function for the output process cannot be obtained directly. Next we consider the tail probability for the maximal delay. Theorem 7 Assume that A(t) (ρ, f) is the (possibly aggregate) input process of a work-conserving system with service rate C > ρ. Then for k ≥ 1, P {D(t) ≥ k} ≤ f (k(C − ρ)).
(17)
PROOF. If, for some j < k, we have Q(t + j) = 0 then we must have D(t) ≤ j < k. Hence, P {D(t) ≥ k} ≤ P {Q(t + j) > 0; j = 0, 1, · · · , k − 1}.
(18)
ˆ C) and notice that when Q(t)(= A(t; ˆ C)) > 0 we have Recall that Q(t) = A(t; ˆ ρ) ≥ A(t; ˆ C) + C − ρ. Furthermore, if Q(t) > 0 and Q(t + 1) > 0, we have A(t; ˆ + 1; ρ) =A(t; ˆ ρ) + A(t + 1) − ρ A(t b C) + C − ρ + A(t + 1) − ρ ≥A(t;
b C) + A(t + 1) − C + 2(C − ρ) =A(t; b + 1; C) + 2(C − ρ). =A(t
Inductively, if Q(t) > 0, Q(t + 1) > 0, · · · , Q(t + k − 1) > 0, then we have ˆ + k − 1; ρ) ≥ A(t ˆ + k − 1; C) + k(C − ρ). A(t
(19)
Therefore, by (18) and (19), ˆ + k − 1; ρ) > k(C − ρ)} ≤ f (k(C − ρ)). P {D(t) ≥ k} ≤ P {A(t
5.2
First In First Out (FIFO)
In this part, we assume that the system adopts the FIFO service discipline. If, at time t, Q(t−1) < C, we assume that upon service, ties are broken arbitrarily 13
among arrivals that arrive at t simultaneously. In this case, from Theorem 5, it is known that for the whole aggregate, its output process A(out)(t) is gSBB with A(out)(t) ≤ A(t) (ρ, g) where ρ and g are as shown by equations (14) and (15). However, for each individual source, we cannot obtain much better result than Theorem 6 for its output process. For maximal delay, due to FIFO, each individual source has the same performance as the whole aggregate. And, for the aggregate, obviously, we have &
'
Q(t) D(t) = , C where d·e is the ceiling function denoting the smallest integer which is greater than or equal to a given number. Then D(t) > k is equivalent to Q(t) > kC. ˆ C) ≤ A(t; ˆ ρ) + ρ − C. Hence, Since C > ρ, when Q(t) > 0 we have Q(t) = A(t; for k ≥ 1, ˆ ρ) > kC − ρ} ≤ g(kC − ρ). P {D(t) ≥ k} = P {Q(t) > (k − 1)C} ≤ P {A(t; In summary, we have the following theorem: Theorem 8 Assume that N sources Aj (t) (ρj , fj ) pass through a workP conserving system with capacity C > ρ = N j=1 ρj and FIFO discipline. Then, the maximal delay for each source and that for the whole aggregate A(t) = PN j=1 Aj (t) satisfy, for any k ≥ 1, P {D(t) ≥ k} ≤ g (kC − ρ) .
(20)
where g is as shown by equation (15). Remark. It is interesting to see from Theorems 7 and 8 that FIFO can have better performance in terms of maximal delay for each individual source than a general scheduler whose service discipline is unknown.
5.3
Strict Priority (SP)
We assume that if i < j then the source i has a higher priority than the source j, which means that source j will not be served if there exists workload from source i waiting for service. Within the same source we adopt FIFO discipline. We shall consider the characterization of the output traffic and probabilistic upper bound on delay for each source under the SP discipline. For A1(t), it is exactly the same as it passes through a work-conserving server (out) with capacity C > ρ1 . By Theorem 5, we see that A1 (t) (ρ1 , f1 ). For A2(t), comparing to other traffic streams, A1(t)+A2(t) has the highest priority. 14
Hence it is equivalent to the case that A1 (t) + A2(t) passes through a workconserving server with capacity C > ρ1 + ρ2 . By Theorem 6 we get that (out) A2 (t) (ρ2, [g,2 ]1). By similar discussions we can get g3 , · · · , gN , and the following result. Theorem 9 Suppose that we have N sources sharing a work-conserving system with service rate C, which serves according to strict priority order. Let Aj (t) be the input process from source j. Assume that Aj (t) (ρj , fj ), where PN (out) (out) (t) be the output process of source j. Then A1 (t) j=1 ρj < C. Let Aj (out) (ρ1 , f1) and for j ≥ 2, we have Aj (t) (ρj , [g,j ]1 ) for any > 0, with 1 g,j (x) = gj (x) + ε where
g (x) 2
Z
∞ x
gj (u) du
= [f1 (p21 x) + 2f2 (p22x)]1
···
gN (x)
=
hP N
i=1
for any pki ∈ (0, 1) satisfying
fi (pN i x) + fN (pN N x)
Pk
i=1
i
(21)
1
pki + pkk = 1, k = 2, · · · , N.
Similarly, we can get the following result for delay: Theorem 10 Use Dj (t) to denote the maximal delay of source j. With the same assumption of Theorem 9, we have, for k ≥ 1, P {D1 (t) ≥ k} ≤ f1 (kC − ρ1 )
(22)
P {Dj (t) ≥ k} ≤ gj (k(C − rj )),
(23)
and for j ≥ 2, where rj = ρ1 + · · · + ρj and gj (x) =
j X fi (pji x) i=1
with pji > 0 and
5.4
Pj
i=1
(24)
1
pji = 1.
Generalized Processor Sharing (GPS)
Assume that we have a work-conserving server with capacity C shared by N flows using the GPS service discipline [32]. Assign the i-th flow a parameter φi > 0. Same as in the above, use Q(t) to denote the total queue length at time t and Qi (t) to denote the portion of the queue which belongs to flow i. 15
For GPS, if in the time interval (s, t], Qi (τ ) > 0, then, for any s < τ ≤ t [32], (out)
Ai
(s, t)
(out) Aj (s, t)
≥
φi , j = 1, 2, · · · , N. φj
(25)
P
Without loss of generality, we assume that N i=1 φi = 1 and call φi the i-th serving weight. By (25) we see that if traffic from source i is backlogged all the time in the interval (s, t], then the available serving rate for source i is at least φi C for the whole time period. Theorem 11 Assume that we have N sources sharing a work-conserving system with service rate C which adopts GPS service discipline. For 1 ≤ i ≤ N , we assign source i with a sharing weight φi > 0. Assume that Aj (t) (ρj , fj ), P (out) (t) where Aj (t) is the input process from source j, and N j=1 ρj < C. Let Aj (out) be the output process of source j. Then, Aj (t) (ρj , gj ) where if φj C > ρj , gj (x) = fj (x); otherwise, gj (x) is determined by Theorem 6. If φj C > ρj , Theorem 11 can be proved in the same way as Theorem 5. If we do not have φj C > ρj , of course, we can apply Theorem 6. For maximal delay, since under GPS, if Qj (t) > 0, the amount of work waiting in queue Qj (t) will be served at a rate not less than φj C, we can then obtain the following result from Theorem 8: Theorem 12 Use Dj (t) to denote the maximal delay of source j. With the same assumption of Theorem 11, if φj C > ρj , we have, for k ≥ 1, P {Dj (t) ≥ k} ≤ fj (kφj C − ρj ).
(26)
Remark. Note that when some other conditions on the sources and the weight assignment are satisfied in the GPS system, improved results could be obtained for each source. For example, in an early work of one of the authors, the upper bounds on individual queue length distribution in GPS with long range dependent traffic inputs were studied [39].
6
Traffic with gSBB
Having introduced the definition and some useful properties for gSBB, we present in this section its stochastic ordering monotonicity property, with which we further show that many well-known types of traffic can be represented using gSBB. 16
6.1
Stochastic Ordering Monotonicity of gSBB
As defined in (5), Aˆ (t; ρ) = sups≤t (A (s, t) − ρ · (t − s)) , which can be interpreted as the length of backlog in queue in a virtual constant rate server (with rate ρ) fed with traffic A. We hence expect sups≤t {A(s, t) − ρ · (s, t)} to have similar properties of queue length in a G/G/1 system [36]. Particularly, assuming that A(t) is stationary and ergodic and E{A(1)} < ρ to satisfy the Loynes’ stability condition [29], we then have the following stochastic ordering monotonicity of gSBB, whose proof follows a similar approach as for the monotonicity property of queue length distribution (see e.g. [36] [5]): Theorem 13 (Monotonicity) Suppose A(t) is stationary and ergodic. Then, if E{A(1)} < ρ for some ρ > 0, we have for all t ≥ 0, ˆ Aˆ (t; ρ) ≤st Aˆ (t + 1; ρ) ≤st · · · ≤st A(∞; ρ), ˆ ˆ ρ) as t → ∞. where A(∞; ρ) denotes the steady state of A(t; We shall see in the next subsection that many types of traffic have this kind of bounding function. In such cases, Theoremo 13 tells that if the traffic of a n ˆ ρ) > x at any time t is always upperflow is stationary and ergodic, P A(t; n o ˆ ρ) > x , bounded by the steady-state queue length distribution, i.e. P A(∞; in the virtual SSQ system that has constant service rate ρ and the same traffic input. It also implies that if the steady-state queue length distribution in a SSQ is known for a certain type of traffic, then, with Theorem 13 and the definition of gSBB, it is clear that this type of traffic has gSBB with bounding function given by the steady-state queue length distribution.
6.2
Traffic with gSBB
In the literature, many well-known or widely studied traffic models assume stationary and ergodic arrival process. Consequently, for such types of traffic, Theorem 13 provides a convenient way to find their gSBB representation. In the remaining, we review some such types of traffic and give their gSBB bounding functions.
6.2.1 Poisson traffic We begin with Poisson traffic. Suppose each packet of a flow has the same size L and packet arrivals of the flow follow a Poisson process with mean arrival rate λ. Then, based on Fry’s state equations for M/D/1 [11], it is straightforward 17
to get the steady-state queue length distribution, from which we can conclude that the flow is gSBB with A(t) hρ, f P oisson i for any ρ > λL, where, with a = λL/ρ and k = d Lx e, f
P oisson
(x) = 1 − (1 − a)
k X
"
i=0
#
[a(i − k)]i −a(i−k) e . i!
(27)
6.2.2 Gaussian arrival process We next consider Gaussian arrival process. Let rˆt and vˆ(t) respectively be the mean and variance of the Gaussian arrival process A(t). Then, available simulation and analytical results in the literature [7] [23] [1][30] suggest that (x+(ρ−ˆ r)s)2 ˆ r) > x}, for all ρ > rˆ, exp −inf s≥0 2ˆv(s) is an approximation of P {A(ρ; and “may in fact be a general upper bound, but no proof of this is known” [30]. Hence, for this type of traffic, we may say it is gSBB with A(t) hρ, f Gaussian i with ! (x + (ρ − rˆ)s)2 Gaussian f (x) = exp − , (28) 2ˆ v∗ where vˆ∗ ≡ vˆ(s∗ ) and s∗ is chosen such that at s∗.
(x+(ρ−ˆ r)s)2 2ˆ v (s)
reaches its minimum
6.2.3 (σ(θ), ρ(θ))-upper constrained traffic A stochastic traffic model (σ(θ), ρ(θ)), called θ-MER (minimum envelope rate with respect to θ), has been proposed for studying stationary and ergodic traffic [5][6]. It is shown that many types of traffic can be represented using this (σ(θ), ρ(θ)) model [5][6], which include exponential On-Off, Markov Modulated Process (MMP) and traffic whose effective bandwidth exists. Specifically, a flow is said to be (σ(θ), ρ(θ))-upper constrained for some θ(> 0), if for all 0 ≤ s ≤ t, 1θ log Eeθ(A(t)−A(s)) ≤ ρ(θ)(t − s) + σ(θ). Using Lemma 3.7 in [5], it ˆ ρ) > x} ≤ β(θ)e−θx. Hence, is obtained that if ρ(θ) < ρ, then for all t, P {A(t; if a flow is (σ(θ), ρ(θ))-constrained, then it is gSBB with A(t) hρ, f M ERi with all ρ > ρ(θ) and (29) f M ER(x) = β(θ)e−θx where β(θ) = eθσ(θ) (1 − eθ(ρ(θ)−r) )−1 . An immediate implication is that exponential On-Off and Markov Modulated Process (MMP) types of traffic can be readily represented using gSBB and their corresponding bounding functions can be found from their (σ(θ), ρ(θ)) characterizations [5][6]. In addition, if the effective bandwidth function of a traffic flow is known, its gSBB characterization is also easily obtained from its MER representation [5][6] and (29). 18
6.2.4 SBB For traffic modeled by SBB with upper rate r and bounding function f ∈ F, if its arrivals are independent of each other and satisfy certain conditions, Theorem 2.(ii) implies that it is gSBB with the same bounding function. If, however, the arrival A(t) does not meet the conditions of Theorem 2.(ii), then, from Theorem 3.(ii), P {Q(t; ρ) > x} ≤ f (x) + 1 f (1) (x) for ρ > r. In other words, if f (1)(x) exists, the traffic is also gSBB with A(t) hρ, f SBB i for any > 0, with ρ = r + , and 1 f SBB (x) = f (x) + f (1) (x).
(30)
6.2.5 α−stable traffic It is worth highlighting that the traffic models reviewed above have bounding functions with exponential forms and/or belonging to the function set F for which, any f ∈ F implies its n−fold integration also belongs to F . However, some traffic models may not have bounding functions in F . One example is the α−stable traffic model [12][21][22]. The α−stable is a traffic model characterizing the self-similar behavior of traffic [21][22]. The model is defined by four parameters: (α, H, c1 , c2). In [21], it is shown that the queue length of a constant rate server fed with α−stable ρ−m −α ˆ , where ρ denotes the rate of traffic satisfies P {A(t; ρ) > x} ≤ Cα c1 the server, and Cα and m are parameters determined from (α, H, c1 , c2 ) and x. Then, such α−stable traffic may also be modeled using gSBB with A(t) hρ, f α i where r − m −α f α (x) = Cα . (31) c1 6.3
Relationship with m.b.c Stochastic Arrival Curve and Global Effective Envelope
In a recent work, the maximum-(virtual)-backlog-centric (m.b.c) stochastic arrival curve model is proposed for stochastic traffic modeling [18]. The following is a generalized definition [20]: Definition 3 A flow is said to have a θ–m.b.c stochastic arrival curve α(t) with respect to θ, with bounding function f θ (x), denoted by A(t) ∼θ−mb hf θ , αi, iff for all t ≥ 0 and all x ≥ 0, there holds P { sup [ sup (A(u, s) − α(s − u)) − θ · (t − s)] > x} ≤ f θ (x), 0≤s≤t 0≤u≤s
19
(32)
where θ is some non-negative real value, α(t) is non-negative non-decreasing on t, and for any given θ > 0, f θ (x) is non-negative and non-increasing on x. Letting θ = 0 in (32), Definition 3 is reduced to the model introduced in [18]. The following theorem establishes the relationship between gSBB and the θ– m.b.c stochastic arrival curve model. Its proof is included in the appendix. Theorem 14 If a flow has a θ–m.b.c stochastic arrival curve α(t) = ρ · t with bounding function f θ (x), it is gSBB with A(t) hρ, f θ i. Conversely, if a flow is gSBB with A(t) hρ, fi and the bounding function satisfies f ∈ F, it has a θ–m.b.c arrival icurve ρ · t with bounding function f θ (x), where h stochastic R f θ (x) = f (x) + 1θ x∞ f (y)dy , for any θ > 0. 1
Through the above relationship, interesting properties and results of the θ– m.b.c stochastic arrival curve model [18][20] can be readily borrowed to the gSBB model, particularly when bounding functions are in F . In [3], the global effective envelope model is introduced, which takes advantage of the notion of empirical envelope or minimum envelope process (MEP) introduced in [5]. The MEP of a flow A(t) is defined as A(t) = sup A(s, s + t).
(33)
s≥0
A flow is said to have a global effective envelope g iff there has [3] P {A(t) ≤ g (t)} ≥ 1 −
(34)
where ε is a constant that may be interpreted as the targeted violation probability. To make the expression of the global effective envelope model consistent with SBB and gSBB, we suppose below = f (x) and g (t) = ρ · t + f −1 () in (34) and then have the following form for global effective envelope: P {A(t) − ρ · t > x} ≤ f (x).
(35)
Since we always have A(s, t) ≤ A(t − s) by the definition of MEP, (35) implies that if a flow has a global effective envelope, it belongs to SBB with upper rate ρ and bounding function f (·) if f is in F . Then, based on the relationship between SBB and gSBB found in Theorem 3, the flow also has gSBB with upper rate ρ + θ and bounding function f (x) + 1θ f (1)(x) for any θ > 0. This result can also be easily obtained from the following relationship between global effective envelope and m.b.c stochastic arrival curve: Theorem 15 If a flow has a global effective envelope with upper rate ρ and bounding function f ∈ F , it has a m.b.c arrivali curve ρ · t with h stochastic R bounding function f δ (x), where f δ (x) = f (x) + 1δ x∞ f (y)dy , for any δ > 0. 1
20
Theorem 3 implies that gSBB is more general than SBB in the sense that if a traffic source can be modeled using SBB, it can also be modeled using gSBB. As implied by (33) and also discussed in [3], the condition for a flow to have global effective envelope is more stringent than to have (local) effective envelope or equivalently SBB. Hence, gSBB is also more general than global effective envelope. Theorem 14 implies a larger bounding function is generally needed for θ–m.b.c stochastic arrival curve characterization than for gSBB, which further implies that an analysis based on θ–m.b.c stochastic arrival curve will result in looser bounds or less accurate results. These relationships assert that gSBB is possibly a better traffic model for stochastic service guarantee analysis. More discussion on the relationships between the various traffic models can be found from [20].
7
Conclusions
In this paper we have introduced the concept of generalized Stochastically Bounded Burstiness (gSBB) for modeling traffic, which is based on the virtual backlog property of an input process. We have also studied the superposition of gSBB traffic and the input-output relation. The input-output relation for a constant rate server under the gSBB model tells that the output traffic can be less bursty than the input, which, however, cannot be directly concluded under other traffic models including EBB, SBB and (σ(θ), ρ(θ)). More importantly, we have studied a work-conserving system shared by a number of gSBB sources, to analyze the properties of each output traffic and the bounding probabilities for maximal delay under various service disciplines. Finally, we introduced a stochastic ordering monotonicity property of gSBB. With this property, it can be shown that many well-known traffic models can be readily represented using the proposed gSBB model. These results set up the basis for a fundamental calculus for gSBB traffic. The concept of gSBB has been used to study conformance deterioration in networks with service level agreements (e.g. see [26]) and isolation of long range dependence in generalized processor sharing (e.g. see [39]). In addition, it has been generalized to a model called virtual-backlog-centric (v.b.c) stochastic arrival curve [19] and many interesting results can be found (e.g. see [20]). Since gSBB (as well as its generalization v.b.c stochastic arrival curve) has less restriction in the definition than other traffic models used in stochastic network calculus, we expect that the results in this paper will be of particular interest in the analysis and provision of stochastic service guarantees. This work leaves many open issues for further research. For example, more general server models may be used [27] [18] [20], the analysis may be extended to the network case [17], and the bounding function may be time-dependent. 21
Appendix: Proofs
Proof of Theorem 2
The first part holds trivially, since it is always true that A(s, t) − ρ · (t − s) ≤ ˆ ρ). sups≤t {A(s, t) − ρ · (t − s)} = A(t; For the second part, the proof is based on a known result for stochastic ordering. Suppose that for random variables {X(1), . . . , X(t)} and {Y (1), . . . , Y (t)}, there holds {X(1), . . . , X(t)} ≤st {Y (1), . . . , Y (t)}. Then, for the mapping Z(t) = Φ(X1 , . . . , Xt ), if it is non-decreasing in {x1, . . . , xt}, one has Z 0 (t) ≤st Z 00(t) where Z 0 (t) = Φ(X(1), . . . , X(t)) and Z 00 = Φ(Y (1), . . . , Y (t)) (see e.g. Theorem 2.2.4 in [36] or Theorem 4.3.3 in [31]). ˆ ρ) = sups≤t {A(s, t)−ρ·(t−s)} = max(A(0, t)− For our proof, it is known A(t; + ρ · t, . . . , A(t − 1, t) − ρ) . Let the mapping be Z(t) = Φ(X1 , . . . , Xt ) = max(Xt , . . . , X1 )+ that is clearly non-decreasing. With the given condition, ˆ ρ) ≤st max(X, . . . , X)+ = X and the second part is proved. we obtain A(t;
Proof of Theorem 14
The first part follows by setting s = t on the left hand side of (32). For the second part, it is known: sup [ sup (A(u, s) − (s − u)ρ) − θ · (t − s)] ≤ sup 0≤s≤t 0≤u≤s
n
ˆ ρ) − θ · (t − s) A(s;
o+
0≤s≤t
ˆ ρ) = sup where A(s; 0≤u≤s [A(u, s) − (s − u)ρ]. ˆ ρ) > x} ≤ f (x), there holds for any x ≥ 0, Since P {A(s;
P
(
h
ˆ ρ) − θ · (t − s) sup A(s;
i+
)
>x
0≤s≤t
≤
t X
≤
s=0 ∞ X u=0
n
o
ˆ ρ) − θ · (t − s) > x ≤ P A(s; f (x + θ · u) ≤ f (x) +
1 θ
Z
t X
f (x + θ · (t − s))
s=0 ∞
f (y)dy x
with which, the second part is easily verified. 22
(36)
Proof of Theorem 15
By definition, we have for any s ≥ u ≥ 0, A(s − u) = supv≥0 A(v, s − u + v) ≥ A(u, s). Then, there holds: sup [ sup (A(u, s) − (s − u)ρ)] 0≤s≤t 0≤u≤s
≤ sup [ sup (A(s − u) − (s − u)ρ)] = sup [ sup (A(v) − ρ · v)] 0≤s≤t 0≤u≤s
0≤s≤t 0≤v≤s
= sup (A(v) − ρ · v)
(37)
0≤v≤t
with which and following the same approach as in proving Theorem 3.(ii) as well as in proving Theorem 14 above, we get P {sup0≤v≤t (A(v) − ρ · v) > x} ≤ f (x) + 1δ f (1)(x) for any δ > 0 and hence the theorem follows from (37).
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