Statistical Multiplexing Gain of Link Scheduling Algorithms in QoS Networks (Short Version)

Transcription

1 1 Statistical Multilexing Gain of Link Scheduling Algorithms in QoS Networks (Short Version) Technical Reort: University of Virginia, CS-99-23, July 1999 Robert Boorstyn Almut Burchard Jörg Liebeherr y Chaiwat Oottamakorn Deartment of Electrical Engineering, Polytechnic University, Brooklyn, NY Deartment of Mathematics, University of Virginia, Charlottesville, VA y Deartment of Comuter Science, University of Virginia, Charlottesville, VA Abstract A statistical network service which allows a certain fraction of traffic to not meet its QoS guarantees can extract additional caacity from a network by exloiting statistical roerties of traffic. Here we consider a statistical service which assumes statistical indeendence of flows, but does not make any assumtions on the statistics of traffic sources, other than that they are regulated, e.g., by a leaky bucket. Under these conditions, we resent functions, socalled local effective enveloes and global effective enveloes, which are, with high certainty, uer bounds of multilexed traffic. We show that these enveloes can be used to obtain bounds on the amount of traffic on a link that can be rovisioned with statistical QoS. A key advantage of our bounds is that they can be alied with a variety of scheduling algorithms. In fact, we show that one can reuse existing admission control functions that are available for scheduling algorithms with a deterministic service. We resent numerical examles which comare the number of flows with statistical QoS guarantees that can be admitted with our effective enveloe aroach to those achieved with existing methods. This reort is an abbreviated version of [1]. I. INTRODUCTION Performance guarantees in QoS networks are either deterministic or statistical. A deterministic service guarantees that all ackets from a flow satisfy given worst-case end-to-end delay bounds and no ackets are droed in the network [2], [4], [8], [15]. A deterministic service rovides the highest level of QoS guarantees, however, it leaves a significant ortion of network resources on the average unused [22]. A statistical service makes robabilistic service guaran- This work is suorted in art by the National Science Foundation through grants NCR (CAREER), ANI , and DMS , and by the New York State Center for Advanced Technology in Telecommunications (CATT). tees, for examle, of the form: Pr[Delay > X] <" or Pr[Loss] <": By allowing a fraction of traffic to violate its QoS guarantees, one can imrove the statistical multilexing gain at network links and increase the achievable link utilization. The key assumtion that leads to the definition of statistical services is that traffic arrivals are viewed as random rocesses. With this assumtion a statistical service can imrove uon a deterministic service by (1) taking advantage of knowledge about the statistics of traffic sources, and (2) by taking advantage of the statistical indeendence of flows. Since it is often not feasible to obtain a reliable statistical characterization of traffic sources, recent research on statistical QoS has attemted to exloit statistical multilexing without assuming a secific source model. Starting with the seminal work in [8], researchers have investigated the statistical multilexing gain by only assuming that flows are statistically indeendent, and that traffic from each flow is constrained by a deterministic regulator, e.g., by a leaky bucket [5], [8], [7], [9], [10], [12], [16], [17], [19], [20], [21]. Henceforth, we will refer to traffic which satisfies these assumtions as regulated adversarial traffic. In this aer we attemt to rovide new insights into the roblem of determining the multilexing gain of statistically indeendent, regulated, but otherwise arbitrary traffic flows at a network link. We introduce the notion of effective enveloes, which are, with high certainty, uer bounds on the aggregate traffic of regulated flows. We use effective enveloes to devise admission control tests for a statistical service for a large class of scheduling algorithms. We show that with effective enveloes, admission control for a statistical service can be done in a similar

2 2 flow 1 flow N (unregulated) arrivals... Α 1 (τ)... Α N (τ) Regulators A 1 (t, t+ τ) A N (t, t+ τ) (regulated) arrivals Buffer with Scheduler Fig. 1. Regulators and Scheduler at a Link. fashion as with deterministic enveloes for a deterministic service [2], [4]. In fact, we show that one can reuse admission control conditions derived for various acket scheduling algorithms in the context of a deterministic service, e.g., [4], [15], [23]. Note that only few results are available on statistical multilexing of adversarial traffic, which can consider scheduling algorithms other than a simle multilexer [7], [12]. Related work to this aer are all attemts to consolidate the deterministic network calculus [4] with statistical multilexing (e.g., [2], [6], [10], [11], [12], [14]). In addition, of articular relevance to this aer are all revious results on statistical multilexing gain with adversarial regulated traffic, as cited above. We refer to [1] for a detailed discussion of related work. The results derived in this aer only aly to a single node. Since traffic from multile flows assing through the same sequence of congested nodes may become correlated, the assumtion of statistical indeendence of flows may not hold in such a setting. Only few results are currently available on end-to-end QoS guarantees for adversarial regulated traffic [7], [20], [21]. The remaining sections of this aer are structured as follows. In Section II we secify our assumtions on the traffic and define the effective enveloes. In Section III we derive sufficient schedulability conditions for a general class of acket schedulers, which can be used for a deterministic and (two tyes of) statistical QoS guarantees. In Section IV, we use large deviations results to derive bounds for effective enveloes. In Section V we comare the statistical multilexing gain attainable with the effective enveloes aroach to those obtained with other methods ([8], [12], [19]). In Section VI we resent conclusions of our work. II. TRAFFIC ARRIVALS AND ENVELOPE FUNCTIONS We consider traffic arrivals to a single link with transmission rate C. As shown in Figure 1, the arrivals from each flow are oliced by a regulator, and then inserted into a buffer. A scheduler determines the order in which traffic in the buffer is transmitted. In the following, we view traffic mainly as continuous-time fluid-flow traffic. Note, however, that our discussion alies, without restrictions, C to discrete-time or discrete-size (acketized) views of traffic arrivals. QoS guarantees for a flow j are secified in terms of a delay bound d j. A QoS violation occurs if traffic from flow j exeriences a delay exceeding d j. (We assume that delays consist only of waiting time in the buffer and transmission time.) A. Traffic Arrivals Traffic arrivals to the link come from a set of flows which is artitioned into Q classes C q, each containing N q flows. (Each flow may itself be an aggregate of the traffic from multile sessions.) The traffic arrivals from flow j in an interval [t 1 ;t 2 ) are denoted as A j (t 1 ;t 2 ). We assume that a traffic flow is characterized by a family of random variables A j (t 1 ;t 2 ) which is characterized as follows: (A1) Additivity. For any t 1 < t 2 < t 3, we have A j (t 1 ;t 2 )+A j (t 2 ;t 3 ) = A j (t 1 ;t 3 ). (A2) Subadditive Bounds. Traffic A j is regulated by a deterministic subadditive enveloe A as j A j (t; t + ) A j ( ) 8t 0; 8 0 : (1) (A3) Stationarity. The A j are stationary random variables, i.e., 8t; t 0 > 0 Pr[A j (t; t + ) x] =Pr[A j (t 0 ;t 0 + ) x] : (2) In other words, all time shifts of A j are equally robable. (A4) Indeendence. The A i and A j are stochastically indeendent for all i 6= j. (A5) Homogeneity within a Class. Flows in the same class have identical deterministic enveloes and identical delay bounds. So, A i = A and d j i = d j if i and j are in the same class. Henceforth, we denote by d q the delay bound associated with traffic from class q. By A Cq we denote the arrivals from class q, thatis,a Cq (t; t + ) = P j2c q A j (t; t + ). Remarks: We want to oint out that the above assumtions are quite general. The class of subadditive deterministic traffic enveloes is the most general class of traffic regulators [4], [2]. The assumtions on the randomness of flows are also quite general. Note that, different from [9], [10], we do not require ergodicity. The traffic regulators most commonly used in ractice are leaky buckets with a eak rate enforcer. Here, traffic on flow j is characterized by three arameters (P j ; j ; j ) with a deterministic enveloe given by A j ( ) = min fp j; j + j g 8 0 ; (3)

3 3 where P j j is the eak traffic rate, j is the average traffic rate, and j is a burst size arameter. We will use this tye of regulators in our numerical examles in Section V. A consequence of subadditivity of the A j is that the limit j := lim!1 A j ( )= exists, and that it rovides an uer bound for the longterm arrival rate for A j. We will assume without loss of generality, that for all t, A j (t; t + ) lim = j : (4)!1 B. Definition of Effective Enveloes We next define local effective enveloes and global effective enveloes which are, with high certainty, uer bounds on aggregate traffic from a given class q. The enveloes will be defined for a set of flows C with arrival functions A j and aggregate traffic A C (t; t + ) = P j2c A j(t; t + ). Definition 1: A local effective enveloe for A C (t; t+ ) is a function G C ( ; ") that satisfies for all 0 and all t Pr A C (t; t + ) G C ( ; ") 1, ": (5) In other words, a local effective enveloe rovides a bound for the aggregate arrivals A C (t; t + ) for any secific ( local ) time interval of length. Under the stationarity assumtion (A3), Eqn. (5) holds for all times t, rovided that it only holds for one value t = t o. It is easy to see that there exists a smallest local effective enveloe, since the minimum of two local effective enveloes is again such an enveloe. Note, however, that local effective enveloes are in general not subadditive in, but satisfy the weaker roerty G C ( ;" 1 + " 2 ) G C ( 1 ;" 1 )+G C ( 2 ;" 2 ) : (6) A local effective enveloe G C ( ; ") is a bound for the traffic arrivals in an arbitrary, but fixed interval of length. Global effective enveloes, to be defined next, are bounds for the arrivals in all subintervals [t; t + ) of a larger interval. For the definition of global effective enveloes, we take advantage of the notion of emirical enveloes, as used in [2], [22]. Consider a time interval I of length. The emirical enveloe E C ( ; ) of a collection C of flows is the maximum traffic in subintervals of I as follows: E C ( ; ) = su A C (t; t + ) : (7) [t;t+ )I Definition 2: A global effective enveloe for an interval I of length is a subadditive function H C ( ; ) which satisfies Pr E C ( ; ) H C ( ; ;"); 80 1, ": (8) The attribute global is justified since H C ( ; ;") is a bound for traffic for all intervals of length in I. Now, due to stationarity of the A j, Eqn. (8) holds for all intervals of length, if it holds for one secific interval I. When alied to scheduling, we will select such that it has at least the length of the longest busy eriod. 1 Assuming that one has obtained local or global effective enveloes searately for each traffic class, the following lemma hels to obtain bounds for the traffic from all classes. Lemma 1: Given a set of flows that is artitioned into Q classes C q, with arrival functions A Cq.LetG Cq and H Cq be local and global effective enveloes for class q. Then the following P inequalities hold. (a) If q G C q (;") x, then, for all t, P i Prh A q C q (t; t + ) >x <Q ". P (b) If H C q (;; ") x( ) for all,then h P i Pr 9 : E C q (;) >x( ) <Q ". The rather simle roof of the lemma can be found in [1]. Our derivations in Section IV will make it clear that for " small enough, neither G Cq nor H Cq are very sensitive with resect to ", so that the bounds for " and Q " are comarable. III. DETERMINISTIC AND STATISTICAL SCHEDULABILITY CONDITIONS In this section, we resent three schedulability conditions for a general class of work-conserving scheduling algorithms. The first condition, exressed in terms of deterministic enveloes, ensures deterministic guarantees. The second and third conditions, which use the local and global effective enveloes, resectively, yield statistical guarantees. All three schedulability conditions will be derived from the same exression for the delay of a traffic arrival in an arbitrary work-conserving scheduler (Eqn. (14) in Section III-A). In our discussions, we will not take into consideration that acket transmissions on a link cannot be reemted. This assumtion is reasonable when acket transmission times are short. For the secific scheduling algorithms considered in this aer, accounting for nonreemtiveness of ackets does not introduce rincial 1 For arrival functions A j and regulators with deterministic enveloes A j, the longest busy eriod in a work-conserving scheduler is given by: inff > 0; ( ) g. P j2c A j

4 4 difficulties, however, it requires additional notation (see [15]). Also, to kee notation minimal, we assume that the transmission rate of the link is normalized, that is C =1. A. Schedulability Suose a (tagged) arrival from a flow j in class q (j 2C q ) arrives to a work-conserving scheduler at time t. Without loss of generality we assume that the scheduler is emty at time 0. We will derive a condition that must hold so that the arrival does not violate its delay bound d q. Let us use A q;t (t 1 ;t 2 ) to denote the traffic arrivals in the time interval [t 1 ;t 2 ) which will be served before a class q arrival at time t. LetA q;t C (t 1 ;t 2 ) denote the traffic arrivals from flows in C which contribute to A q;t (t 1 ;t 2 ). Suose that t, is the last time before t when the scheduler does not contain traffic that will be transmitted before the tagged arrival from class q. Thatis, = inffx 0 j A q;t (t, x; t) xg : (9) So, in the time interval [t, ;t) the scheduler is continuously transmitting traffic which will be served before the tagged arrival. (Note that is a function of t and q. To kee notation simle, we do not make the deendence exlicit.) Given, the tagged class-q arrival at time t will leave the scheduler at time t + if > 0 is such that = inf out j A q;t (t, ;t + out ) + out : (10) Hence, the tagged class-q arrival does not violate its delay bound d q if and only if 8 9 out d q : A q;t (t, ;t + out ) + out : (11) Then, the traffic arrival does not have a deadline violation if d q is selected such that su A q;t (t, ;t + d q ), d q : (12) In general, Eqn. (12) is a sufficient condition for meeting a delay bound. For FIFO and EDF schedulers, the condition is also necessary [15]. 2 For a secific work-conserving scheduling algorithm, let (with, d q ) denote the smallest values for which A C (t, ;t+ ) A q;t C (t, ;t + d q ) : (13) 2 A FIFO scheduler transmits traffic in the order of arrival times. An EDF (Earliest-Deadline-First) scheduler tags traffic with a deadline which is set to the arrival time lus the delay bound d q, and transmits traffic in the order of deadlines. Remark: For most work-conserving schedulers one can easily find such that equality holds in Eqn. (13). For examle, for FIFO, SP, 3 and EDF schedulers, we have: 8 < : FIFO: =0, ;>q SP: = 0 ; = q d q ;<q EDF: = maxf,;d q, d g With Eqn. (13), the arrival from class q at time t does not have a violation if d q is selected such that su ( X A C (t, ;t + ), ) d q : (14) Next, we show how Eqn. (14) can be used to derive schedulability conditions for deterministic and statistical services, using deterministic enveloes, local effective enveloes, and global effective enveloes. For a deterministic service, the delay bound d q must be chosen such that Eqn. (14) is never violated. For a statistical service, d q is chosen such that a violation of Eqn. (14) is a rare event. B. Schedulability with Deterministic Enveloes Exloiting the roerty of deterministic enveloes in Eqn. (1), we can relax Eqn. (14) to su 8 < X X : 9 = A ( j + ), j2c ; d q : (15) Since, + is not deendent on t, we have obtained a sufficient schedulability condition for an arbitrary traffic arrival. We refer the reader [15] to verify that for FIFO and EDF scheduling algorithms the condition in Eqn. (15) is also necessary, in the sense that if it is violated, then there exist arrival atterns conforming with A j leading to deadline violations for class q. For SP scheduling, the condition is necessary only if the deterministic enveloes are concave functions. Next we resent bounds on the likelihood of a violation of Eqn. (14), using local and global effective enveloes. C. Schedulability with Local Effective Enveloes With Eqn. (14), the robability that the tagged arrival from time t exeriences a deadline violation is less than " 3 An SP (Static Priority) scheduler assigns each class a riority level (we assume that a lower class index indicates a higher riority), and has one FIFO queue for traffic arrivals from each class. SP always transmits traffic from the highest riority FIFO queue which has a backlog.

5 5 if d q is selected such that Pr " su ( X A C (t, ;t + ), ) d q # 1, ": (16) Let us, for the moment, make the convenient assumtion that Pr " su ( X " X su Pr ) d q # A C (t, ;t + ), # A C (t, ;t + ), d q :(17) Assuming that equality holds in Eqn. (17), we can re-write Eqn. (16) as su Pr " X A C (t, ;t + ), d q # 1, ": (18) Remark: The assumtion in Eqn. (17) requires further justification, since, in general, the right hand side is larger than the left hand side. On the other hand, several works on statistical QoS have used Eqn. (17) with equality [3], [11], [12], [13], [14], and, in several cases, have suorted the assumtion with numerical examles. Recall from the definition of the local effective enveloe that G C (;") x imlies Pr A C (t; t + ) >x < ". Then, with Lemma 1(a) and assuming that Eqn. (17) holds with equality, we have that a class-q arrival has a deadline violation with robability <"if d q is selected such that ( ) X su G C ( +; "=Q), d q : (19) With Eqn. (19) we have found an exression for the robability that an arbitrary traffic arrival results in a violation of delay bounds. This condition can be viewed as a general formulation of the schedulability conditions for statistical QoS from [11], [12], [14]. The drawback of the condition in Eqn. (19) is its deendence on the assumtion in Eqn. (17). Emirical evidence from numerical examles, including those resented in this aer, as well as numerical evidence from revious work which emloyed this assumtion [3], [12], suggests that Eqn. (19) is not overly otimistic. However, it should be noted that the bound in Eqn. (19) is not a rigorous one. D. Schedulability with Global Effective Enveloes We next use global effective enveloes to exress the robability of a deadline violation in a time interval. We will see that this bound, while more essimistic, can be made rigorous. Consider again the traffic arrival from class q which occurs at time t. The arrival time t lies in a busy eriod of the scheduler I of length at most, which starts at time t, and which ends at a time after the tagged arrival has dearted. Using the roerties of the emirical enveloe E C,as defined in Section II, we have that, for all t and + 0, E C ( + ; ) A C (t, ;t + ): (20) Thus, we can only have a deadline violation if 9 : ( X E C ( + ; ), ) d q : (21) With Lemma 1(b), the robability that an arrival from class q exeriences a deadline violation in the interval I is <",ifd q is selected such that su ( X H C ( + ; ; "=Q), ) d q : (22) Note that the nature of the statistical guarantees derived with local effective enveloes (in Subsection III-C) and with global effective enveloes (in Subsection III-D) are quite different. Local effective enveloes are (under the assumtion in Eqn. (17)) concerned with the robability that a articular traffic arrival results in a deadline violation. Global effective enveloes address the robability that a deadline violation occurs for some arrival in a certain time interval. Clearly, a service which guarantees the latter is more stringent, and will lead to more conservative admission control. Lastly, we want to oint to the structural similarities of the conditions in Eqs. (15), (19), and (22). Thus, schedulability conditions which have been derived for a deterministic service can be reused, without modification, for a statistical service if effective enveloes are available. IV. CONSTRUCTION OF EFFECTIVE ENVELOPES In this section we will construct the local and global effective enveloes G C and H C for the aggregate traffic from a set of flows as described in (A1)-(A5). Throughout this section, we will work only with flows from a single class. So, we will dro the index q ; and C and N, resectively, will denote the set of flows and the number of flows. We

6 6 denote by A ( ) the common deterministic enveloe for the flows in C, and by A C (t; t + ) the aggregate traffic. Our derivations roceed in the following stes: Ste 1. We comute bounds for the moments of the individual flows A j (t; t + ). Since the flows are indeendent, this directly leads to bounds for the moments of A C (t; t + ). 4 Ste 2. We use the Chernoff bound to determine a local effective enveloe G C directly from our bounds on the moments. Ste 3. We use a geometric argument to construct H C from any local effective enveloe G C. Secifically, we will rovide bounds of the following nature: G C ( ; ") H C ( ; ;") G C ( 0 ; " 0 ) : (23) where 0 = > 1 and " 0 =" < 1 deend on. We claim that for " sufficiently small and not too large, 0 = 1, and resulting global effective enveloe is reasonably close to the local effective enveloe. A. Moment bounds The moment generating functions of the distributions of A C and the A j are defined as follows: M C (s; ) := E[e A C (t;t+ )s ] ; (24) M j (s; ) := E[e A j (t;t+ )s ] : (25) Due to the stochastic indeendence of the flows, we can write: M C (s; ) = NY j=1 M j (s; ) : (26) Thus, to obtain a bound on M C (s; ), it is sufficient to bound the moment-generating function of a single flow A j (t; t + ). The following lemma rovides such a bound. We refer to [1] for a roof. Lemma 2: Assume that A(t; t + ) satisfies Conditions (A1), (A2), and (A3). Then, M (s; ) 1+ A ( ) e sa ( ), 1 : (27) Combining Eqn. (26) with (27) of Lemma 2 yields the bound M C (s; ) 1+ A ( ), e sa ( ), 1 N : (28) 4 Note that the moment generating function for arrival functions A j is also comuted in [2]. However, different from [2], our arrivals A j are regulated by deterministic functions A j. B. Local Effective Enveloes B.1 Using the Central Limit Theorem The bound in Eqn (28) can be strengthened to bounds for individual moments. A case of articular interest is the bound for the variance N (A ( {z ), } ) ; (29) =:~s 2 =:s 2 Var[A C (t; t + )] {z } where we have used the bound on the second moment together with the assumtion that E [A C (t; t + )] =. An alication of the Central Limit Theorem, will now yield a bound which is equivalent to Knightly s bound on the rate variance in [12]. Using first the Central Limit Theorem and then the bound on the variance in Eqn.(29), we see that for x> Pr[A C (t; t + ) Nx] Nx, N 1, ~s N x, 1, s (30) ; (31) where is the cumulative normal distribution. Here, ~s and s, resectively, are the square roots of the left hand and right hand sides of Eqn. (29). To find G C so that Pr[A C (0;) G C ( ; ")] "; (32) we set Pr[A C (t; t + ) Nx] " in Eqn. (31) and solve for Nx. This gives us an (aroximate) local effective enveloe as G C ( ; ") N + z N where z s A ( ), 1 ; (33) j log (2")j is defined by 1, (z) =". B.2 Using the Chernoff Bound While the estimate in Eqn. (33) is asymtotically correct, for finite values of N it is only an aroximation. To obtain a rigorous uer bound on Pr[A C (0;) Nx],recall the Chernoff bound for a random variable Y [18]: Pr[Y y] e,sy E[e sy ] 8s 0 : (34) In articular, for A C,thisgives Pr[A C (0;) Nx] e,nxs M C (s; ) (35) e,xs 1+, N e sa ( ) A, 1 (36) : ( )

7 7 Here, Eqn. (35) simly used the Chernoff bound, and Eqn. (36) used Eqn. (28). Since we have a choice for selecting s in Eqn. (36), we want to make the bound as small as ossible. For x<a ( ), the right hand side is minimal when s is chosen so that e sa ( ) = x A ( ), A ( ), x : (37) Substituting this value of s into Eqn. (36) yields Pr A C (0;) Nx x x A () A ( ), A ( ), x 1, x A N () (38) : Again, our goal is to find G C satisfying Eqn. (32). Using the bound in Eqn. (38) and enforcing that G C ( ; ") is never larger than NA ( ) we may set G C ( ; ") = N min(x; A ( )) ; (39) where x is set to be the smallest number satisfying the inequality x A () A ( ), 1, x A () x A " 1=N : (40) ( ), x It can be verified that for N sufficiently large, this bound matches closely the CLT bound of Eqn. (33). Remark: For deterministic enveloes with a eakrate constraint A ( ) P, both exressions for G C in Eqn. (39) and Eqn. (33) describe lines, with sloes which deend on, P, N, and". In other words, the arrivals A C (t; t + ) satisfy, with robability at least 1, ", again a rate constraint. The new rate differs from the mean rate N by an error of order N (for fixed values of, P,and "). C. From Local to Global Effective Enveloes We use the results from the revious subsection to construct a global effective enveloe H C for A C. The first ste is a geometric estimate for E C for a articular value of in terms of the local effective enveloe. The second ste fixes the value of the global effective enveloe for a finite collection of values i. Finally, we obtain the entire enveloe by extraolation. Let us define two events: B(x; t; ) = fa C (t; t + ) Nxg : (41) B (x; ) = fe C ( ; ) Nxg : (42) for an arbitrary interval I of length. The event B(x; t; ) occurs if the arrivals in the secific time interval [t; t + ] exceed Nx, while B (x; ) occurs if there τ / k Interval of length Ι β τ τ' Fig. 2. Embedding Intervals. is some interval of length in the interval I where the arrivals exceed Nx. With Eqn. (38), we have a bound for the robability of events B(x; t; ). The following bound for B (x; ) in terms of B(x; t; ) will be used to construct H C (; ;") from G C (; "). Lemma 3: Let k 2 be a ositive integer, I an interval of length, t 2 I, and 0. Then Pr[B(x; t; )] Pr[B (x; )] k Pr[B(x; t; 0 )] ; (43) with 0 = =(k +1)=k. Proof: By stationarity, we may assume that I = [0;] and t = 0. The left inequality holds by definition, since B(x; 0;) B (x; ). To see the inequality on the right, let t i = i=k (i = 0;:::;dk=e), and consider the intervals [t i ;t i+k+1 ] of length 0 = k+1 for k i = 1;:::;d(, )k=e (all but ossibly the last are subintervals of [0;].) See Figure 2 for an illustration of this construction. Clearly, every subinterval of length in I is contained in at least one of the intervals of length 0. The claim now follows with stationarity. 2 Lemma 3 rovides a bound on arrivals in all subintervals of length in I. One of its imlications is that for every value of, Pr E C ( ; ) G C, k +1 k ; " k "; (44) where E C is the emirical enveloe, and G C is any local effective enveloe. We next assign a finite number of values for H C ( ; ;"): Pick a collection of values i and k i (i = 1;:::;n) and define H C ( i ; ;") = G C ( 0 i ; "0 ) ; (45)

8 8 where 0 i = k +1 k i and " 0 = " nx i=1 k i i!,1 : (46) To justify this construction, note that by Eqn. (44) we have Pr h 9i : E C ( i ; ) G C ( 0 i;" 0 ) i nx i=1 k i i " 0 (47) ": (48) To get values for the global effective enveloe on intervals ( i,1 ; i ) and [0; 1 ), we first extraolate, using the bound A and monotonicity, and then enforce subadditivity. More recisely, we set H C ( ; ;") = Pinf i = X where f is an auxiliary function defined by f ( ) = 8 < : i f ( i ) : (49) min fh C ( i,1 ; ;")+A (, i,1 ); H C ( i ; ;")g 2 [ i,1 ; i );i =2;:::;n min fa ( ); H C ( 1 ; ;")g 2 [0; 1 ) (50) In other words, H C is the largest subadditive function which does not exceed f. Since there exists no universal best global effective enveloe, it is clearly imossible to make an otimal choice for the values of i and k i. It is, however, ossible to make good choices, which lead to global effective enveloes that aroximate the given local effective enveloe well, at least when " is sufficiently small. In our numerical results, we use k i = k; i = i o (i =1;:::;n) ; (51) where o is a small number, and we choose =1+ 1 k +1 ; k z z + N ; (52) s where z is defined by 1, (z) = " and s by Eqn. (29). The choice of the i in Eqn. (51) guarantees that, k +1 H C ( ; ;") G C ;" 0 ; (53) k for all 2 [ o ;], where, by Eqn. (46), " 0 = o(, 1) ": (54) k In [1] we rovide a justification for the choice of k and in Eqn. (52) for eak-rate constrained traffic with large burst sizes. This is done with a heuristic otimization which alies the CLT aroximation from Eqn. (33). V. EVALUATION In this section, we evaluate the effective enveloe aroach, using the schedulability conditions from Section III and the bounds derived in Section IV. The key criteria for evaluation is the amount of traffic on a link which can be rovisioned with QoS guarantees. As benchmarks for statistical QoS rovisioning we consider the following non-statistical methods: Peak Rate: Peak rate allocation, rovides deterministic QoS guarantees, but, is an inefficient method for achieving QoS. Deterministic: We use admission control tests for deterministic QoS from Eqn. (15). The admissible traffic varies with the scheduling algorithm. Average Rate: Average rate allocation only guarantees finiteness of delays and average throughut. We will evaluate the two methods for rovisioning statistical QoS which are resented in this aer. Local Effective Enveloe: Here we use Eqn. (18) to determine : admissibility. We will evaluate the quality of the following two bounds, derived in Section IV: Local Effective Enveloe (CB): Uses the bound from Eqn. (40), obtained with the Chernoff bound. Local Effective Enveloe (CLT): Uses the bound from Eqn. (33), obtained with the Central Limit Theorem. Recall from our discussion in Section IV that the local effective enveloe (CLT) results are equivalent to the ratevariance enveloe method described in [12]. Global Effective Enveloe: We use Eqn. (22) to determine admissibility. The global effective enveloe is constructed by first finding (see Footnote 1), and choosing a number o which is small comared to the delay bounds. We determine the arameters, k,and i according to (51) and (52). We then aly Eqn. (45) for each of the i,and comlete the rocess by the extraolation in Eqs. (49) and (50). (In Eqn. (45), we use the local effective enveloe (CB) rather than the corresonding CLT bound, since the latter would yield only aroximate bounds.) We comare our results with the effective bandwidth aroach for regulated adversarial traffic from the literature: Effective Bandwidth [8], [16], [19]: 5 The effective bandwidth aroach assigns to each flow a fixed caacity, the effective bandwidth, and assumes that each flow is serviced at a rate which corresonds to the effective bandwidth. The delay bounds will be indirectly derived from the buffer size. We set the delay bound d to d = B=C, whereb is 5 The cited works calculate effective bandwidth for regulated adversarial sources. The comlete literature on effective bandwidth is much more extensive.

9 9 the buffer size at the scheduler and C is the transmission rate of the link. In our examles, we include the following results on effective bandwidth: EB-EMW: This is the result from the classical aer by Elwalid/Mitra/Wentworth (Eqn. (39) in [8]). EB-RRR: We use Eqn. (9) from [19] by Rajagoal/Reisslein/Ross which resents an imrovement to the EB-EMW result. In all our exeriments, we consider traffic regulators which are obtained from eak rate controlled leaky buckets with deterministic enveloes as given in Eqn. (3). 6 In all exeriments, we consider a link with C = 45 Mbs, and we consider two traffic classes. The traffic arameters of a flow in one of the classes are as follows: Class Peak Rate Mean Rate Burst Size P (Mbs) (Mbs) (bits) To arameters are selected so as to match, at least aroximately, the examles resented in [8], [19]. We will resent three sets of examles. In the first examle, we comare the deterministic enveloes with our bounds for the local and global effective enveloes for sets of homogeneous sources. In the second examle, we comare the maximum number of admissible flows in a FIFO scheduler for a given delay bound d and delay-violation robability ". In the third examle, we investigate the case of heterogeneous traffic with different QoS requirements, and we comare the admissible regions for different scheduling algorithms (SP, EDF). A. Examle 1: Comarison of Enveloe Functions In the first examle, we study the shae of local and global effective enveloes for homogeneous sets of flows, as functions of the lengths of time intervals. The enveloes are comared to the deterministic enveloe A ( j ) = minfp j ; j + j ;g, to the eak rate function P j,and to the average rate function j. In our grahs, we lot the amount of traffic P er flow for the various enveloes (e.g., we resent j2c G j( ; ")=N). Figures 3(a) and 3(b) show the results for multilexed flows from Class 1 and Class 2, resectively. We set " = 10,6 for all enveloes. By deicting the amount of traffic er flow for different numbers of flows (N denotes the number of flows), we can observe how the statistical multilexing gain increases with the number of flows. 6 Most of the methods listed here can work with more comlex regulators. However, since eak-rate enforced leaky buckets are widely used in ractice, they serve as good benchmarks. The first observation to be made is that the local and global effective enveloes are much smaller than the deterministic enveloe or the eak rate. Another observation is that, for a fixed number of flows N, the global effective enveloe is larger than the local effective enveloes, and the local effective enveloe bound is smaller when using CLT (central limit theorem), as comared to CB (Chernoff bound). Note, however, that bounds for the enveloes with CLT are only aroximate, and may be too otimistic, esecially for small number of flows. Figure 3 also shows that local and global effective enveloes converge as the number of flows N is increased. B. Examle 2: Admissible Region for Homogeneous Flows In this examle, we investigate the number of flows admitted by various admission control methods for guaranteeing QoS at a link with a FIFO scheduler. We assume that flows are homogeneous, that is, all flows belong to a single class. Again, the robability of a violation of QoS guarantees is set to " =10,6. We comare the admissible regions of the local and global effective enveloes, to those of the effective bandwidth techniques (both EB-EMW and EB-RRR), and to a deterministic QoS guarantees. We comare the results with those obtained from a discrete event simulation. For the simulation, we take a attern which we exect, based on the simulations in [17], to be close to an adversarial traffic attern for eak-rate controlled leaky buckets. However, do not claim that the results from the simulation scenario are the worst ossible. In the simulations, the traffic for a flow with arameters given by (P; ; ), has a eriodic attern. A flow transmits at the average rate for a duration T on1 = d=2, whered is the delay bound. Then, the flow transmits at the eak rate P for a duration T on2 = =(P, ), followed by another hase of length T on1 at which the flow transmits at rate. Then, the source shuts off, waits for a duration T of f = = and then reeats the attern. The starting time of a attern of the flows are uniformly and indeendently chosen over the length of its eriod. Figures 4(a) and (b) deict the number of admitted flows as a function of the delay bound. The figures show that all methods for statistical QoS admit many more connections than a deterministic admission control test. In both Figures, the effective enveloes (both CLT and CB) are closest to the simulation results. (Once again, we oint out that the results using the local effective (CLT) bounds are identical to the rate-variance results resented in [12].) Note, however, that results obtained with local effective enveloes are aroximate and are not guaranteed to be

10 10 Amount of traffic er flow (kbits) Peak Rate Deterministic Enveloe Global Effective Enveloe Local Effective Enveloe (CB) Local Effective Enveloe (CLT) N = N = Average Rate Time interval (ms) (a) Class 1. N = 1000 Amount of traffic er flow (kbits) Peak Rate Deterministic Enveloe Global Effective Enveloe Local Effective Enveloe (CB) Local Effective Enveloe (CLT) N = N = 1000 N = Average Rate Time interval (ms) (b) Class 2. Fig. 3. Examle 1: Comarison of Enveloe Functions for 100 ms, " = 10,6, and for Number of Flows N = 100; 1000; Number of admissible connections Average Rate Fig. 4. Simulation Local Eff. Env. (CLT) Local Eff. Env. (CB) 0.75 Global Eff. Env. Bufferless MUX EB-RRR EB-EMW Deterministic Peak Rate Delay bound (ms) (a) Class Average utilization Number of admissible connections Simulation Local Eff. Env. (CLT) Local Eff. Env. (CB) Peak Rate Delay bound (ms) (b) Class 2. Global Eff. Env. EB-RRR Bufferless MUX EB-EMW Deterministic Average Rate Examle 2: Admissible Number of Connections at a FIFO Scheduler for Homogeneous Flows as a Function of Delay Bounds (" =10,6, 0 <d 100 ms) Average utilization uer bounds on the admissible regions. Comaring the results from effective enveloes to the effective bandwidth results, we observe that the effective enveloe methods admits more connections than the effective bandwidth methods if delay bounds are large. The difference of the admissible regions in Figure 4(a) to those in Figure 4(b) illustrate the high degree to which the size of the admissible region is deendent on the traffic arameters. The lower burst sizes of flows in Class 2 lead to larger admissible regions for all methods. Secifically, notice that deterministic QoS in Figure 4(b) yields similar results to the statistical methods, if the delay bounds are large. C. Examle 3: Admissible Region for Heterogeneous Traffic Here we investigate an examle with different scheduling algorithms and with heterogeneous traffic arrivals. As scheduling algorithms, we consider Static Priority (SP) and Earliest-Deadline-First (EDF). For a deterministic service, EDF is otimal, in the sense that the admissible regions with EDF scheduling is maximal [15]. To our knowledge, results for a statistical service (with adversarial traffic), have not been reorted for EDF. In this examle, we multilex a number of flows from Class 1 and from Class 2 on 45 Mbs. We fix the delay bounds, such that the delay bound for Class-1 flows is relatively long, d 1 = 100 ms, and the delay bound for Class-2 flows is relatively short, d 2 = 10 ms. For any articular method, we determine the maximum number of Class-1 and Class-2 flows that can be suorted simultaneously on the 45 Mbs link. The result are shown in Figure 5. The lot deicts the admissible region for SP and EDF scheduling, using the results for the (two tyes of) local effective enveloes, effective enveloes, and deterministic enveloes. We also include the admissible regions for the effective bandwidth aroaches (EB-EMW and EB-RRR). Note, however, that

11 11 Number of Class-1 connections Average Rate Local Eff. Env (CLT) Local Eff. Env. (CB) Global Eff. Env. Deterministic For Local and Global Eff. Env. : Thick lines = EDF scheduling Thin lines = SP scheduling 50 Bufferless MUX EB-RRR Peak Rate 0 EB-EMW Number of Class-2 connections Fig. 5. Examle 3: Admissible Region of Multilexing Class 1 and Class 2 Flows with " = 10,6 and d 1 d 2 =10 ms. = 100 ms and the shown effective bandwidth methods assume a simle multilexer (with virtual buffer artitioning) and do not account for different scheduling algorithms. The results in Figure 5 show that the difference between SP and EDF schedulers is small in all cases. The effective enveloe is, again, more conservative than the local effective enveloe method. Finally, Figure 5 illustrates that with heterogeneous flows and the effective bandwidth methods (EB-EMW, EB-RRR) may not erform as well as methods which consider scheduling algorithms. We also erformed a simulation for the EDF scheduling algorithm. For the simulations, we used a source model which was shown to be adversarial for a simle multilexer with buffer and bandwidth artitioning [19]. We do not know or claim that this source model is also adversarial for EDF scheduling. However, with this choice, the simulations give the same results as an average rate allocation. VI. CONCLUSIONS We have resented new results on evaluating the statistical multilexing gain for acket scheduling algorithms. A useful roerty of our aroach is that it searates the consideration of the service definition (deterministic, statistical), the scheduling algorithm (FIFO, SP, EDF), and the mathematical methodology (Central Limit Theorem, Chernoff Bound). Thus, our work may be useful to researchers who want to determine the statistical multilexing gain for other traffic regulators, scheduling algorithms, or large deviation results. 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