A General Theory of Computational Scalability Based on Rational Functions

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1 A General Theory of Comutational Scalability Based on Rational Functions Neil J. Gunther arxiv: v1 [cs.pf] 11 Aug 2008 August 25, 2008 Abstract The universal scalability law (USL) of comutational caacity is a rational function C = P ()/Q() with P () a linear olynomial and Q() a second-degree olynomial in the number of hysical rocessors, that has been long used for statistical modeling and rediction of comuter system erformance. We rove that C is equivalent to the synchronous throughut bound for a machine-reairman with state-deendent service rate. Simler rational functions, such as Amdahl s law and Gustafson seedu, are corollaries of this queue-theoretic bound. C is both necessary and sufficient for modeling all ractical characteristics of comutational scalability. 1 Introduction For several decades, a class of real functions called rational functions [1], has been used to reresent throughut scalability as a function of hysical rocessor configuration. In articular, Amdahl s law [2], its modification due to Gustafson [3] and the Universal Scalability Law (USL) [4] have found ubiquitous alication. In this context, the relative comuting caacity, C, is a rational function of the number of hysical rocessors. It is defined as the quotient of a olynomial P () in the numerator and Q() in the denominator, i.e., C = P ()/Q(). Each of the above-mentioned scalability models is distinguished by the number of coefficients or fitting arameters associated with the olynomials in P () and Q(). For examle, Amdahl s law and Gustafson s modification are single arameter models, whereas the USL model contains two arameters. Desite their historical utility, these models have stood in isolation without any deeer hysical interretation. Is has have been suggested that Amdahl s law is not fundamental [5]. More imortantly, the lack of a unified hysical interretation has led to the use of certain flawed scalability models [6]. In this note, Performance Dynamics Comany, 4061 East Castro Valley Blvd., Suite 110, Castro Valley, CA 94552, USA. nj erfdynamics. com 1

2 we demonstrate that the aforementioned class of rational functions corresonds to certain erformance bounds belonging to a queue-theoretic model. The idea that Amdahl s law, which has most frequently been associated with the scalability of massively arallel systems, can be considered from a queuetheoretic standoint, is not entirely new [See e.g., 7, 8]. However, quite aart from motivations entirely different from our own, those revious works emloyed oen queueing models with an unbounded number of requests (See Aendix C), whereas we shall use a closed queueing model with a finite number of requests corresonding to the number of hysical rocessors. The USL function is associated with a state-deendent generalization of the machine reairman [9]. The organization of this aer is as follows. We briefly review the scalability models of interest in Sect. 2. The aroriate queueing metrics associated with the standard machine reairman and its state-deendent extension are discussed in Sect. 3. The erformance characteristics associated with synchronous queueing are also resented there. The main theorem (Theorem 2) is established in Sect. 4. Amdahl s law and Gustafson s linear seedu are shown to be corollaries of this theorem. Finally, in Sect. 5 we rove an earlier conjecture that a rational function with Q() a second-degree olynomial is both necessary and sufficient to model all ractical cases of comutational scalability. 2 Parametric Models Although technically, we are discussing rational functions, we shall hereafter refer to them as arametric models, and the coefficients as arameters, since the rimary alication of these models is nonlinear statistical regression of erformance data [See e.g., 4, 10, 11, 12, and references therein]. Definition 1 (Seedu). If an amount of work N is comleted in time T 1 on a unirocessor, the same amount of work can be comleted in time T < T 1 on a -way multirocessor. The seedu S = T 1 /T is one measure of scalability. 2.1 Amdahl s law For a single task that takes time T 1 to execute on a unirocessor ( = 1), Amdahl s law [2] states that if the task can be equiartitioned onto rocessors, but contains an irreducible fraction of sequential work σ [0, 1], then only the remaining ortion of the execution time (1 σ)t 1 can be executed as arallel subtasks on hysical rocessors. The bound on the achievable equiartitioned seedu [13] is given by the ratio which simlifies to T 1 S (σ) = ( (1) 1 σ σt 1 + )T 1 S (σ) = 1 + σ( 1) ; (2) 2

3 Figure 1: Parametric models: USL (red), Amdahl (green), Gustafson (blue), with arameter values exaggerated to distinguish their tyical characteristic relative to ideal linear scaling (dashed). The horizontal line is the Amdahl asymtote at σ 1. a rational function with P () = and Q() a first-degree olynomial. As the rocessor configuration is increased, i.e.,, the number of concomitantly smaller subtasks also increases and the seedu (1) aroaches an asymtote, shown as the horizontal in Fig Gustafson s seedu S (σ) σ 1, (3) Amdahl s law assumes the size of the work is fixed. Gustafson s modification is based on the idea of scaling u the size of the work to match. This assumtion results in the theoretical recovery of linear seedu S G (σ) = σ + (1 σ) (4) Equation (4) is a rational function with Q() = 1 and P () a first-degree olynomial in. Although (4) has insired various efforts for imroving arallel rocessing efficiencies, achieving truly linear seedu has turned out to be difficult in ractice. Most recently, (4) has been roosed as a way to otimize the throughut of multicore rocessors [14]. Definition 2 (Relative Caacity). As an alternative to the seedu, scalability can also be defined as the relative caacity, C = X()/X(1), where X() is the throughut with rocessors, and X(1) the throughut of a single rocessor. 3

4 Table 1: Alication domains for the USL model A: Ideal concurrency (σ, κ = 0) B: Contention-limited (σ > 0, κ = 0) Single-threaded tasks Tasks requiring locking or sequencing Parallel text search Message-assing rotocols Read-only queries Polling rotocols (e.g., hyervisors) C: Coherency-limited (σ = 0, κ > 0) D: Worst case (σ, κ > 0) SMP cache inging Tasks acting on shared-writable data Incoherent alication state between Online reservation systems cluster nodes Udating database records 2.3 Universal Scalability Law (USL) The USL model [4, 10, 11] is a rational function with P () = and Q() a second-degree olynomial: C (σ, κ) = 1 + σ( 1) + κ ( 1) where the coefficients belonging to the terms in the denominator have been regroued into three terms with two arameters (σ, κ). These terms can be interreted as reresenting: 1. Ideal concurrency associated with linear scalability (σ, κ = 0) 2. Contention-limited scalability due to serialization or queueing (σ > 0, κ = 0) 3. Coherency-limited scalability due to inconsistent coies of data (σ, κ > 0) Table 1 summarized how these arameter values can be used to classify the scalability of different tyes of alications. Equation (5) subsumes (2) and (4). In articular, (2) is identical to (5) with κ = 0. The key distinction is that, unlike (2), (5) ossesses a maximum at 1 σ = (6) κ which is controlled by the arameter values according to: (a) 0 as κ (b) as κ 0 (c) κ 1/2 as σ 0 (d) 0 as σ 1 The imortant imlication is that beyond the throughut becomes retrograde. See Fig. 1. This effect is commonly observed in alications that involve sharedwritable data (Case D in Table 1). In the subsequent sections, we attemt to rovide deeer insight into the hysical significance of (5) by recognizing its association with queueing theory. (5) 4

5 3 Queueing Models The machine reairman (Fig. 2) is a well-known queueing model [15] which reresents an assembly line comrising a finite number of machines which break down after a mean lifetime Z. A reairman takes a mean time S to reair a broken machine and if multile machines fail, the additional machines must queue for service in FIFO order. The queue-theoretic notation, M/M/1//, imlies exonentially distributed lifetimes and service eriods with a finite oulation of requests and buffering., Z R() X() S Figure 2: Conventional machine reairman queueing model comrising machines with mean utime Z (to) and a reair queue (bottom) with mean service time S. 3.1 Queueing Metrics The erformance characteristics of interest for the subsequent discussion are the throughut X() and residence time R(). Definition 3 (Throughut). The throughut, X = N/T, is the number of tasks N comleted in time T. Definition 4 (Residence Time). The mean residence time R = W +S is the sum of the time sent waiting for service W lus the actual reair time S when the reairman services the machine. On average, the number of machines that are u is ZX, while Q are down (for reairs), such that the total number of machines in either state is given by = Q + ZX. Rearranging this exression roduces: Q = ZX and alying Little s law [15] (Q = XR) [15] XR = ZX 5

6 Table 2: Interretation of the queueing metrics in Fig. 2 Interretation Metric Reairman Multirocessor Time share machines rocessors users Z u time execution eriod think time S service time transmission time CPU time R() residence time interconnect latency run-queue time X() failure rate bandwidth throughut gives R() = X() Z (7) for the mean residence time at the reair station. Rearranging (7) rovides an exression for the mean throughut of the reairman as a function of : X() = R() + Z Definition 5 (Mean RTT). The denominator in (8), R() + Z, is the mean round-tri time (RTT) for M/M/1//. Since M/M/1// is an abstraction, it can be alied to different comutational contexts. In the subsequent sections, the machines will be taken to reresent hysical rocessors and the time sent at the reair station is taken to reresent the interconnect latency between the rocessors [16, 17]. See Table 2. This choice is merely to conform to the conventions most commonly used in discussions of arallel scalability [11], but the generic nature of queueing model means that any conclusions also hold for software scalability [See e.g., 10, Cha. 6]. 3.2 Synchronous Queueing We consider the secial case of synchronous queueing in M/M/1//. The queuetheoretic erformance metrics defined in Sect. 3.1 are for the steady-state case and therefore each corresonds to the statistical mean of the resective random variable. Moreover, as already mentioned, we cannot give an exlicit exression for R() since its value is deendent on the value of X(), which is also unknown in steady state. Definition 6 (Synchronized Requests). If all the machines in M/M/1// break down simultaneously, the queue length at the reairman is maximized such that the residence time in definition 4 becomes R() = S. This situation corresonds to one machine in service and ( 1) waiting for service and rovides a lower bound on (8) [18, 19, 20]: X() (9) S + Z (8) 6

7 In the context of multirocessor scalability (Table 2), it is tantamount to all rocessors simultaneously issuing requests across the interconnect. Remark 1 (A Paradox). Consider the case where all rocessors have the same deterministic Z eriod. At the end of the first Z eriod, all requests will enqueue at the interconnect (lower ortion of Fig. 2) simultaneously. By definition, however, the requests are serviced serially, so they will return to the arallel execution hase (to ortion of Fig. 2) searately and thereafter will always return to the interconnect at different times. In other words, even if the queueing system starts with synchronized visits to the interconnect, that synchronization is immediately lost after the first tour because it is destroyed by the serial queueing rocess. The resolution of this aradox is discussed in Aendix B. Definition 7 (Synchronous RTT). In the resence of synchronous queueing, the mean RTT of definition 5 becomes S + Z. Lemma 1. The relative caacity C and the seedu S give identical values for the same rocessor configuration. Proof. Let the unirocessor throughut is defined as X 1 = N/T 1 and the multirocessor throughut is X = N/T. Hence follows from definition 1. C = X() X(1) = N T T 1 N S Lemma 2 (Serial Fraction). The serial fraction (2.1) can be exressed in terms of M/M/1// metrics by the identity [6, 10, 11]: { σ = S S + Z 0 as S 0, Z = const., (10) 1 as Z 0, S = const. If σ = 0, then there is no communication between rocessors and the interconnect latency therefore vanishes (maximal execution time). Conversely, if σ = 1, then the execution time vanishes (maximal communication latency). Proof. The RTT for a single (unartitioned) task in Fig. 2 is T 1 = S + Z. The RTT for a equiartitioned subtasks is T = (S/) + Z/. From definition 1, the corresonding seedu is Equating (11) with (1), we find Eliminating T 1 roduces S = S + Z S + Z/ (11) S = σt 1 and Z = (1 σ)t 1. (12) S = which, uon solving for σ, roduces (10). ( ) σ Z (13) 1 σ 7

8 See Aendix B for another ersective. Definition 8. The quantity Z S = 1 σ σ is the service ratio for the M/M/1// model. (14) Theorem 1 (Seedu Duality). Let (σ, π) be a continuous dual-arameter air with σ is the serial fraction (10) and π = 1 σ. The Amdahl seedu (2) is invariant under scalings of (σ, π) by. Proof. Using definition 8, theorem 1 can be reresented diagrammatically as: Z Z/ S S π/σ = Z/S S (σ) Z Z S S (15) The ath on the left hand side of (15) corresonds to reducing the single task execution time by (subtasks) while the interconnect service time remains unchanged. This follows from definition 6, R = S, but the service time for each subtask is also reduced to S/. Hence, R = S. Conversely, the right hand ath of (15) corresonds to tasks, each with unchanged execution time Z, but scaled service time R = S. Both aths result in Amdahl s law (2), which can be seen by first rewriting (11) in terms of the service ratio Z/S (definition 8): S = 1 + Z/S ( ). (16) Z S (a) Interreting the denominator in (16) as belonging to the left hand ath of (15) leads to the exansion S + Z S = S + Z + S S (S + Z)/S = ( ) 1 S + Z + 1 S Collecting terms and simlifying roduces: S = 1 + ( S S + Z ) ( 1) (17) which is identical to (2) uon substituting (10). 8

9 (b) Following the right hand ath in (15) leads to the exansion S = = S(1 + Z/S) S + Z ( ) S + Z S + Z S + Z + S S + Z S S + Z Collecting terms in the denominator roduces: S = ( ) S + 1 S S + Z S + Z (18) which also yields (2) via (10). Remark 2. Theorem 1 anticiates the interretation of Gustafson s law as a consequence of scaling the work size Z Z in M/M/1//. See corollary State-Deendent Service We now consider a generalization of this machine reairman model in which the residence time R includes an additional time that is roortional to the load on the server, exressed as the number of enqueued requests. Since the queue-length is a canonical measure of the state of the system, the reairman becomes a statedeendent server [9, 15], denoted M/G/1//. Let the additional service time be S, then the state-deendent rogression is: = 1 : R = S = 2 : R = 2 S + S = 3 : R = 3 S + 2 S (19) = : R = S + ( 1)S In other words, the extra time sent at the reair station increases linearly with the number down machines. Remark 3. In general, it is exected that S < S. It could, however, be a multile of S, but that is clearly undesirable. Some examle alications of the load-deendent M/G/1// queueing model in a comutational context include: (a) Pairwise Exchange: Modeling the erformance degradation due to combinatoric airwise exchange of data between multirocessor caches or cluster nodes. See Sect. 5. As (19) shows, this effect does not aly to = 1. 9

10 (b) Broadcast Protocol: If any rocessor broadcasts a request for data, the other ( 1) rocessors must sto and resond before comutation can continue [11]. (c) Virtual Memory: Each task is a rogram with its own working set of memory ages. Page relacement relies on a higher latency device, such as a disk. As the number of rograms increases, age relacement latency causes the system to thrash such that the throughut to become retrograde [9]. Cf. Fig Parametric Models as Queueing Bounds In this section, we show that the arametric scalability models in Section 2 corresond to certain throughut bounds on the queueing models in Section 3. Theorem 2 (Main Result). The universal scalability law (5) is equivalent to synchronous relative throughut in M/G/1//. Proof. Let S = cs in (19), with c a ositive constant of roortionality. The residence time for synchronous-requests becomes From definition 2, the scalability function is: R() = S + c( 1)S (20) (S + Z) C = S + c ( 1)S + Z (S + Z) = ( 1)S + (S + Z) + c ( 1)S (S + Z) = (S + Z) [1 + ( 1)S(S + Z) 1 + c ( 1)S(S + Z) 1 ] Collecting terms and simlifying roduces C = 1 + σ( 1) + c σ( 1) (21) (22) where we have alied the σ-identity from lemma 2. Combining the coefficients of the third term in the denominator of (22) as κ = c σ, yields (5). Remark 4. Since c is an arbitrary constant, c > 0 imlies that the arameter κ = c σ in (5) can be unbounded, whereas σ < 1 always. Remark 5. The state-deendence of R() in (20) does not change lemma 2 since σ is determined by S and Z only and both of those queueing metrics are constants. Corollary 1 (Amdahl s law). Amdahl s law (2) is the synchronous bound on relative throughut in M/M/1//. Proof. Follows immediately from the roof of theorem 2 with c = 0 in (21). 10

11 Remark 6. Elsewhere [6, 10, 11], corollary 1 was roven as a searate theorem. Corollary 2 (Gustafson s law). Gustafson s law (4) corresonds to the rescaling Z Z in M/M/1//. Proof. Since Gustafson s result is a modification of Amdahl s law, we start with (21) and let c = 0. Under Z Z, the scalability function becomes: (S + Z) C = S + Z = S + Z S + Z S + (Z + S S) = S + Z Once again, after alication of lemma 2, this simlifies to which is identical to the linear seedu S G in (4). C = σ + σ (23) Remark 7. Rewriting (23) as C = σ/(1 σ) +, we note that the additive constant σ/(1 σ) = S/Z is the inverse of the service ratio in definition 8. In the context of M/M/1//, rescaling the execution time, Z Z, rior to artitioning, adds a fixed overhead (S/Z) to an otherwise linear function of, whereas the overhead in Amdahl s law is an increasing function of. The results of this section have also been confirmed using event-based simulations [12]. 5 Extrema and Universality Ideal linear scalability (C, for > 0) has a ositive and constant derivative. More realistically, large rocessor configurations ( ) are exected to aroach saturation, i.e., the asymtote C sat σ 1. Any hysical comutational system that develos a scalability maximum at in the ositive quadrant ( > 0), means that C must have a negative derivative for > and therefore relative comutational caacity falls below the saturation value or C < C sat, i.e., throughut erformance becomes retrograde. Since this behavior is undesirable, there is little virtue in characterizing the maximum beyond the ability to quantify its location ( ) using a given scalability model. This observation leads to the following conjecture [See also 10,. 65], which we now rove. Conjecture 1 (Universality). For a rational function C = P ()/Q() with P () =, a necessary and sufficient condition for C to be a model of comutational scalability is, Q() = 1 + a 1 + a 2 2 with coefficients a 1, a 2 > 0. The simlest line of roof comes from considering latency rather than throughut. See Section D for further discussion. 11

12 Proof. Ideal latency reduction, T = T 1 /, is a hyerbolic function and therefore has no extrema. The additional latency due to airwise interrocessor communication introduces the combinatoric term, ( 2) = ( 1)/2, such that the total latency becomes T (κ) = T 1 + κt 1 ( 1) (24) 2 with constant κ > 0. Equation(24) has a unique minimum for > 0 (Fig. 3). Substituting (24) into the seedu definition 1, roduces S (κ) = 1 + κ( 1) where we have absorbed the factor of 2 in κ. S (κ) now ossesses a unique maximum in the ositive quadrant ( > 0). Thus, the quadratic term in the denominator of (25) is necessary for the existence of a maximum but it is not sufficient because S (κ) does not exhibit the Amdahl asymtote σ 1 when κ = 0. However, the two-arameter latency T (σ, κ) = T ( ) σt 1 + κ T 1 ( 1) (26) 2 does introduce the required term into (25) and, by lemma 1, is identical to (5). (25) T Figure 3: A minimum occurs in the total latency T () due to an increasing airwiseexchange time being added to the initial latency reduction. The second term in (26) can be interreted as the fixed time it takes any one rocessor to broadcast a request for data and wait for the remaining fraction of rocessors, ( 1)/, to resond simultaneously. This is also another way to view synchronous queueing in M/M/1//. It simly introduces a lower bound, σt 1, on the latency reduction. The third term in (26) is analogous to Brook s law [21]: Adding more manower to a late software roject makes it later, with interreted as eole rather 12

13 than rocessors. Here, for examle, it can be interreted as the latency due to the airwise exchange of data to maintain cache coherency in a multirocessor. 6 Conclusion Several ubiquitous scalability models, viz., Amdahl s law, Gustafson s law and the universal scalability law (USL), belong to a class of rational functions. Treated as arametric models, they are neither ad hoc nor unhysical. Rather, they corresond to certain bounds on the relative throughut of the machine reairman queueing model. In the most general case, the main theorem 2 states that the USL model corresonds to the synchronous throughut bound of a load-deendent machine reairman. USL subsumes both Amdahl s law and Gustafson s law as corollaries of theorem 2. As well as roviding a more hysical basis for these scalability models, the queue-theoretic interretation has ractical significance in that it facilitates rediction of resonse time scalability using (7) and rovides deeer insight into otential erformance tuning oortunities. Aendices A Why Universal? The term universal is intended to convey the notion that the USL (defined in Sect. 2.3) can be alied to any comuter architecture; from multi-core to multi-tier. This follows from the fact that there is nothing in (5) that exlicitly reresents any articular system architecture or interconnect toology. That information is resent but it is encoded in the numeric value of the arameters σ and κ. The same could be said for Amdahl s law but the difference is that, being a rational function with linear Q(), Amdahl s law cannot redict the retrograde scalability commonly observed in erformance evaluation measurements [4, 10]. As roven in Sect. 5, the USL is both necessary and sufficient to model all these effects. The USL does not exclude defining a more general or more comlex scalability model to account for such details as, heterogeneous rocessors or the functional form of degradation beyond, but any such model must include the USL as a limiting case. The best analogy might be to regard the USL as being akin to Newton s universal law of gravitation. Here, universal means generally alicable to any gravitating bodies. Newton s theory has been suerseded by a more sohisticated theory of gravitation; Einstein s general theory of relativity. Einstein s theory, however, does not negate Newton s theory but rather, contains it as a limiting case, when sace-time is flat. Since sace-time is flat in all ractical alications, NASA uses Newton s equations to calculate the flight aths of all its missions. 13

14 B Synchronous Queueing The roofs of theorem 2 (USL) and corollary 1 (Amdahl) emloy mean value equations for metrics which characterize steady state conditions. As noted in remark 1, synchronized queueing cannot be maintained in steady state. Synchronization and steady state are not comatible concets because the former is an instantaneous effect, whereas the analytic solutions we seek are only valid in long-run equilibrium. Elsewhere [10, 12], we have shown that a necessary requirement for maintaining synchronous queueing is to introduce another buffer in addition to the waiting line (W 1 ) at the reairman. If the extra buffer reresents a ost-reair collection oint, such that each reaired machine (comleted request) is held offline until all machines are reaired then, synchronous queueing is maintained rovided the Z eriods are i.i.d. deterministic. The extra buffer acts as a barrier synchronizer. Unfortunately, this is a D/M/1// queue, whereas the roof of corollary 1 is based on M/M/1//. Moreover, the reairman erformance metrics are robust [22], so our results should also hold for G/M/1//. In more recent simulation exeriments [12], we have shown that this restriction on Z eriods can be lifted by ositioning the buffer as a re-reair waiting room. Instead of requiring all machines to break down and enqueue simultaneously, we allow any number, less than, to fail but with the added constraint that when any machine invokes service at the reairman, all other machines (or executing rocessors) must susend oerations as well, i.e., visit the susension buffer. Under these conditions, Z can be G-distributed. Because this intermittent synchronization occurs with much higher frequency and for much shorter average time eriods than barrier synchronization, the otential imact of the G-distributed tails on the Z eriods is truncated. Synchronization can be treated as a two-state Markov rocess, e.g., A: arallel and B: serial, where the B state includes those rocesses that are susended as well as waiting for service. If λ A is the transition rate for A B and λ B for B A, then the robability of being in state B is given by Pr(B) = λ B λ A + λ B (27) In the revious scenario it only takes a single machine to fail to susended all other machines. The failure rate is therefore λ A = 1/Z and the service rate is λ B = 1/S. Substituting these into (27) roduces and exression identical to the serial fraction σ in (10). In state B, some fraction 1 are enqueued and the remainder 2 = 1 are susended. On average, any machine can exect to send time R = ( )S to comlete reairs. Hence, the total serial time is R = S, which is the quantity that aears in the roofs. C Queueing Models of Amdahl s Law Others have also considered Amdahl s law from a queue-theoretic standoint [See, e.g., 7, 8]. Of these, [8] is closest to our discussion, so we briefly summarize the 14

15 differences. First, the motivations are quite different. The author of [8], like many other authors, seeks clever ways to defeat Amdahl s law; in the sense of Gustafson (Sect. 2.2), whereas we are trying to understand Amdahl s law by roviding it with a more fundamental hysical interretation. Ironically, both investigations invoke queue-theoretic models to gain more insight into the ertinent issues; an oen (M/M/m) queue in [8], a closed queue (Fig. 2) in this aer. Second, two stes are undertaken to define an alternative seedu function: 1. An attemt to unify both the Amdahl and Gustafson equations into a single seedu function. 2. Extend that unified seedu function to include waiting times. The overarching goal is to find waiting-time otima for this unified seedu function. The unification ste is achieved by urely algebraic maniulations and does not rest on any queue-theoretic arguments. The oen queueing model rovides an ad hoc means for incororating waiting times as a function of queue length. The subsequent analysis is based entirely on simulation results and thereafter dearts significantly from the analytic aroach of this aer. By virtue of our aroach, we have shown that both Amdahl and Gustafson scaling laws are unified by the same queueing model, viz., the machine-reairman model. Moreover, corollary 1 is a lower bound on throughut; synchronous throughut, and therefore reresents worst-case scalability. With this hysical interretation, it follows immediately that Amdahl s law can be defeated more conveniently than roosed in [8] by simly requiring that all requests be issued asynchronously [12]. D Remarks on the Proof of Conjecture 1 The roof in Section 5, is simlified by using the additive roerties of the latency function T rather than inverting the rational function f() = R()/Q() directly. Since Q() is quadratic in, the temtation is to consider the inverse of f 1 and use the fact that a general quadratic function has a minimum. To see why this aroach runs into difficulties, consider the simlified reresentation of (5): f() = (28) with all unit coefficients. Equation (28) is not invertible in the formal sense because f 1 is not a one-to-one maing, even in the ositive quadrant. We can, however, consider the full inverse with its branches, as shown in Fig. 4. The rincial branch is shown in light blue with the other branches occurring at the extrema of f 1. This corresonds to the extrema of of (28) occurring at = ±1 f(1) = ±1/3. Hence, f 1 C, > 1/3. Alternatively, choosing the denominator be a erfect square: (29) 15

16 Figure 4: The comlete rational function (28) (red) and its inverse (blue). (29) can be exressed either as a roduct of linear factors: ( )( ) = or a artial fraction exansion: = (1 + ) 2 Although such decomositions are suggestive of the need for two arameters (Fig. 5), they would seem to obscure the roof of Theorem 1 rather than illuminate it. Using the latency function T and then inverting it to roduce the corresonding throughut scaling using lemma 1, avoids these roblems. References [1] Wikiedia. Rational function. en.wikiedia.org/wiki/rational function, Aril [2] G. Amdahl. Validity of the single rocessor aroach to achieving large scale comuting caabilities. Proc. AFIPS Conf., 30: , Ar [3] J. L. Gustafson. Reevaluating Amdahl s law. Comm. ACM, 31(5): , [4] N. J. Gunther. A simle caacity model for massively arallel transaction systems. In Proc. CMG Conf., ages , San Diego, California, December

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