Performance Evaluation 64 (2007) 978–993 www.elsevier.com/locate/peva
Bandwidth-sharing networks in overload Regina Egorova a,b,∗ , Sem Borst a,b,c , Bert Zwart d a CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands b Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands c Bell Laboratories, Alcatel-Lucent, P.O. Box 636, Murray Hill, NJ 07974, USA d Georgia Institute of Technology, H. Milton Stewart School of Industrial and Systems Engineering, 765 Ferst Drive, Atlanta, GA 30332-0205, USA
Available online 1 July 2007
Abstract Bandwidth-sharing networks as considered by Massouli´e and Roberts provide a natural modeling framework for describing the dynamic flow-level interaction among elastic data transfers. Under mild assumptions, it has been established that a wide family of so-called α-fair bandwidth-sharing strategies achieve stability in such networks provided that no individual link is overloaded. In the present paper we focus on α-fair bandwidth-sharing networks where the load on one or several of the links exceeds the capacity. Evidently, a well-engineered network should not experience overload, or even approach overload, in normal operating conditions. Yet, even in an adequately provisioned system with a low nominal load, the actual traffic volume may significantly fluctuate over time and exhibit temporary surges. Furthermore, gaining insight into the overload behavior is crucial in analyzing the performance in terms of long delays or low throughputs as caused by large queue build-ups. The way in which such rare events tend to occur, commonly involves a scenario where the system temporarily behaves as if it experiences overload. In order to characterize the overload behavior, we examine the fluid limit, which emerges from a suitably scaled version of the number of flows of the various classes. The convergence of the scaled number of flows to the fluid limit is empirically validated through simulation experiments. Focusing on linear solutions to the fluid-limit equation, we derive a fixed-point equation for the corresponding asymptotic growth rates. It is proved that a fixed-point solution is also a solution to a related strictly concave optimization problem, and hence exists and is unique. We use the fixed-point equation to investigate the impact of the traffic intensities and the variability of the flow sizes on the asymptotic growth rates. The results are illustrated for linear topologies and star networks as two important special cases. Finally, we briefly discuss extensions to models with user impatience. c 2007 Elsevier B.V. All rights reserved.
Keywords: Alpha-fair bandwidth sharing; Flow-level performance; Overload; Fluid model; User impatience; Fixed-point equation
1. Introduction Over the past several years, the processor-sharing discipline has emerged as a useful paradigm for evaluating the flow-level performance of elastic data transfers competing for bandwidth on a single bottleneck link. Bandwidthsharing networks as considered by Massouli´e and Roberts [20,23] provide a natural extension for modeling the dynamic interaction among competing elastic flows that traverse several links along their source–destination paths. ∗ Corresponding author at: CWI, PNA2, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.
E-mail address:
[email protected] (R. Egorova). c 2007 Elsevier B.V. All rights reserved. 0166-5316/$ - see front matter doi:10.1016/j.peva.2007.06.024
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
979
Assuming exponential flow size distributions and Poisson arrivals, De Veciana et al. [24,25] proved that weighted max–min and proportional fair bandwidth-sharing strategies achieve stability in such networks (positive recurrence of the associated Markov process) under the nominal condition that no individual link is overloaded. Bonald and Massouli´e [5] extended that result to a wide family of weighted α-fair bandwidth-sharing strategies as introduced by Mo and Walrand [21]. Massouli´e [19] established that the nominal stability condition remains sufficient for the proportional fair strategy with an additional ‘routing feature’, thus further generalizing the result to phase-type flow size distributions. Bramson [9] showed that the max–min fair strategy guarantees stability under the nominal load condition for general flow size distributions and renewal arrival processes. Under similar general distributional assumptions and load conditions, Gromoll and Williams [13,14] studied the fluid limit for weighted α-fair strategies, and established stability in some special cases, such as linear and tree topologies. In the present paper we focus on α-fair bandwidth-sharing networks where the load on one or several of the links exceeds the capacity. Obviously, with adequate provisioning, a network should not experience overload, or even approach overload, in normal operating conditions. However, even in a properly dimensioned system with a low typical load, the actual traffic volume may substantially fluctuate over time and exhibit transient surges, see also Bonald and Roberts [6]. Furthermore, an understanding of the overload behavior plays a crucial role in analyzing the performance in terms of long transfer delays or low flow throughputs as caused by large queue build-ups. The likely way for such rare events to occur, commonly entails a scenario where the system temporarily appears to deviate from the normal stochastic laws and behaves as if it experiences overload, see for instance Anantharam [2]. As alluded to above, an α-fair bandwidth-sharing network bears a strong resemblance to a single-server processorsharing system, since within each class the bandwidth is fairly shared among all competing flows. The overload behavior of a single-server processor-sharing system was first analyzed by Jean-Marie and Robert [16]. Their analysis was extended by Altman et al. [1] to the discriminatory processor-sharing discipline, which corresponds to a singlenode network with a weighted α-fair strategy. Puha et al. [22] studied a single-server overloaded processor-sharing system in terms of measure-valued processes. There are two key distinctions that arise in a network scenario: (i) the rate received by a class is no longer constant, but depends on the number of flows of all classes in some intricate fashion; and (ii) the network may show non-workconserving behavior due to the fact that congestion at other links may prevent a link from utilizing its full capacity (a phenomenon referred to as entrainment by Kelly and Williams [17]). These two features not only render the flow-level performance largely intractable, but also complicate the analysis of the overload behavior. For example, even on links with excess capacity, the workloads may grow because of the non-work-conserving behavior mentioned above. In addition, while the total number of flows must grow in overload conditions, the exact nature of the growth patterns of the various classes is far from clear, and may even potentially involve oscillatory effects in certain cases as observed in Bramson [8] and Lu and Kumar [18] for example. In order to characterize the growth dynamics, we examine the fluid limit, which emerges when the number of flows of the various classes is scaled in both space and time. The convergence of the scaled number of flows to the fluid limit is empirically validated through simulation experiments. Focusing on linear solutions to the fluid-limit equation, we obtain a fixed-point equation for the coefficients, which represent the corresponding asymptotic growth rates of the queue lengths. It is proved that a fixed-point solution is also a solution to a related strictly concave optimization problem, and therefore exists and is unique. Using the fixed-point characterization, we investigate the impact of the traffic intensities and the variability of the flow sizes on the asymptotic growth rates. The results are illustrated for linear topologies and star networks as two important special cases. While admittedly simple, linear networks provide a useful model for flows that traverse several links and experience bandwidth competition from independent cross-traffic. Star networks offer a convenient abstraction for scenarios where the core is highly over-provisioned and congestion predominantly occurs at the edge with comparatively low-capacity access links, see also Fayolle et al. [10]. Finally, we briefly discuss extensions to models with user impatience, which has a particularly pronounced impact in overload conditions, see also Bonald and Roberts [6]. The remainder of the paper is organized as follows. In Section 2 we present a detailed model description and state some preliminaries. In Section 3 we examine linear solutions to the fluid-limit equation, derive a fixed-point equation for the corresponding asymptotic growth rates, and establish existence and uniqueness of the fixed point. We focus on the special case of a network with a linear topology in Section 4. In Section 5 we turn attention to the special case of a star network. In Section 6 we elaborate on the numerical experiments that we conducted to illustrate and support the analytical findings. We discuss extensions to models with user impatience in Section 7.
980
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
2. Model description and preliminaries In this section we present a detailed model description and state some preliminaries. 2.1. Flow-level model We consider a bandwidth-sharing network as in [20,23] with a finite number of links labeled by j = 1, . . . , J . Denote by C = (C1 , . . . , C J ) the vector of link capacities. The network is offered traffic from several classes indexed by i = 1, . . . , I . Each class is characterized by a route, i.e., a nonempty subset of {1, . . . , J }, which represents the links traversed by the traffic from that class. Let A be a J × I incidence matrix such that A ji = 1 if link j belongs to the route of class i, and A ji = 0 otherwise. For now, we do not make any specific assumptions on the topology of the network or the structure of the route sets. The traffic of the various classes consists of elastic file transfers. A flow is a continuous transfer of a file through the links along the route associated with its class. The duration of the flow thus depends on its size and the simultaneous service rate it receives on all the links along its route. Class-i flows arrive as a renewal process with rate λi , i.e., the mean interarrival time is 1/λi , and have generally distributed sizes Bi with mean 1/µi . Denote by ρ = (ρ1 , . . . , ρ I ) with ρi := λi /µi the vector of traffic intensities. 2.2. Bandwidth-sharing strategy Denote by Λ = (Λ1 , . . . , Λ I ) the vector of rates allocated to the various classes. Any rate allocation vector Λ must satisfy the capacity constraints AΛ ≤ C. The bandwidth arbitration among competing flows is governed by a weighted α-fair strategy [21], which selects a rate allocation vector Λ(z) = (Λ1 (z), . . . , Λ I (z)) based on the population z = (z 1 , . . . , z I ) of active flows and an optimization criterion as specified below. Within each class, the rate is fairly shared among all competing flows, i.e., if z i > 0, then each class-i flow receives service at rate Λi (z)/z i . Thus, a class-i flow that is continuously active throughout the time interval [s, t], receives a cumulative amount of service Z t Λi (Z (u)) du, Si (s, t) = Z i (u) s with Z (t) = (Z 1 (t), . . . , Z I (t)) representing the population of active flows at time t. A weighted α-fair strategy is parameterized by a fairness coefficient α ∈ (0, ∞) and a weight vector w = (w1 , . . . , w I ) ∈ R+ I . For a given population z = (z 1 , . . . , z I ) 6= (0, . . . , 0) of active flows, the weighted α-fair rate allocation Λ(z) is determined by the solution to the optimization problem: (P)
maximize subject to
G z (Λ) AΛ ≤ C, Λ ≥ 0,
I → [−∞, ∞] is defined by where the objective function G z (·) : R+ 1−α I X α Λi , α ∈ (0, ∞)/{1}, w z i i 1−α i=1 G z (Λ) = I X wi z i log Λi , α = 1,
(1)
i=1
with the convention that G z (Λ) = −∞ if α ≥ 1 and Λi = 0, z i > 0. With the additional convention that Λi (z) = 0 when z i = 0, the rate allocation is uniquely determined since the above optimization problem is strictly concave. The family of α-fair bandwidth-sharing strategies includes several common fairness concepts as special cases. In particular, the case α = 1 and the limiting cases α → 0 and α → ∞ correspond to a rate allocation that is proportional fair, achieves maximum throughput, and is max–min fair, respectively.
981
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
2.3. Load conditions Under Markovian assumptions, [5] established that α-fair strategies achieve stability provided no individual link is overloaded, i.e., Aρ < C. In the present paper we focus on an overload scenario where the above condition is violated for at least one of the links. While the total number of flows must grow in such a scenario, the exact nature of the growth patterns of the various classes is not so evident. In the next section we will characterize the growth dynamics in terms of the fluid limit. 3. Main results In this section we present our main results, which characterize the growth rates of the number of flows in an overloaded α-fair bandwidth-sharing network. Specifically, this section examines the fluid limit emerging from a suitably scaled version of the number of flows of the various classes, namely, the sequence of processes (Z (r t)/r, t ≥ 0) with r → ∞. Gromoll and Williams [13] established that the sequence of these scaled processes is tight. In preparation for the statement of the main result, we first introduce a slightly modified version of the rate allocation functions which may be interpreted as the service rates at the fluid scale. The service rate Ri (z) received by class i is defined as follows: Ri (z) ≡ Λi (z) if z i > 0, where Λi (·) is the solution of the optimization problem (P), and Ri (z) ≡ ρi if z i = 0. The above distinction reflects the fact that on the fluid scale, z i = 0 requires that class i receives service at rate ρi , rather than 0. We now proceed to postulate a fluid-model solution (z(t), t ≥ 0). This definition is slightly different from the one in [13]: a non-negative continuous function (z(t), t ≥ 0) is said to be a fluid-model solution if it satisfies Z t P(Bi > Si (s, t))ds, (2) z i (t) = λi 0
where Si (s, t) = s Riz(z(u)) du is the cumulative amount of service received by a class-i flow during the time interval i (u) [s, t]. We refer to Remark 3.2 and the Appendix for a discussion. The next proposition presents the main result of the paper. Rt
Proposition 3.1. The fluid-limit equation (2) admits a linear solution z(t) ≡ mt, where the vector m = (m 1 , . . . , m I ) forms the unique solution to the fixed-point equation mi − R (m) Bi∗ Ri (m) = ρi E e i , i = 1, . . . , I, (3) ∗
and Bi∗ represents a residual flow size (i.e. a random variable with density P(Bi > y)/E[Bi ] and LST E[e−x Bi ] = µi (1 − E[e−x Bi ])/x). The above proposition holds for arbitrary network topologies and arbitrary flow size distributions. In a single-link scenario, i.e., J = 1, it reduces to known results for single-server processor-sharing type systems. In particular, in the single-class case, i.e., I = 1, we have, dropping the class index, R(m) = 1, and Eq. (3) specializes to ∗
1 = ρ E[e−m B ], which corresponds to the result in [16]. In the multi-class case, we have Ri (m) = wi m i / takes the form I P −wi−1 wk m k Bi∗ wi m i k=1 , i = 1, . . . , I, = ρi E e I P wk m k
PI
k=1 wk m k ,
and Eq. (3)
k=1
which agrees with the result in [1] for overloaded discriminatory processor-sharing queues. We will give additional examples for specific network topologies in Sections 4 and 5. The remainder of the section is organized as follows. We first provide a heuristic interpretation of the fixed-point equation (3). Next, we give the proof of Proposition 3.1. Finally, we discuss some qualitative properties of the growth rates.
982
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
3.1. Heuristic interpretation The fixed-point equation (3) may be heuristically derived in a similar way as explained by Jean-Marie [15]; we are not aware of an article where this derivation has been published. Suppose that Z r(r ) → m a.s. as r → ∞ for some . Let vector m = (m 1 , . . . , m I ). Then, for large t, a class-i flow will receive service at a rate of approximately Rim(mt) it ain be the arrival epoch of the nth class-i flow. Then the size Bi of that flow and its sojourn time Ti may be related as: Z Bi =
ain +Ti
ain
Ri (mu) du. mi u
Since Ri (mu) = Ri (m) (see [17]), it follows that Z a n +Ti i mi 1 Bi = du = log(ain + Ti ) − log ain . n Ri (m) u ai Taking the exponent on both sides, we obtain m i B Ti = ain e Ri (m) i − 1 . The number of active class-i flows at time t may then be expressed as Z i (t) = #{n : ain + Tin ≥ t, ain ≤ t} = #{n : −R
mi Bi i (m)
}. Because ain ≈ n/λi for large n, we have mi mi Bi − R (m) Bi − R (m) i i ≈ λi t E 1 − e . Z i (t) = # n : t ≥ n/λi ≥ te
t ≥ ain ≥ te
Dividing both sides by t and letting t tend to infinity, we deduce mi Bi − R (m) i , m i = λi 1 − E e
(4)
which is equivalent to Eq. (3). Remark 3.1. In the case of exponential flow sizes, Eq. (3) specializes to m i = λi − µi Ri (m),
i = 1, . . . , I,
(5)
which makes sense, since µi Ri (m) is indeed the departure rate of class-i flows. This is also consistent with the convention Ri (m) = ρi when m i = 0. Remark 3.2. Note that we have not established convergence of (Z (r t)/r, t ≥ 0) to (z(t), t ≥ 0). In order to prove convergence, one would need to show that (i) any convergent subsequence of (Z (r t)/r, t ≥ 0) is a fluid-model solution in our sense; and (ii) that our fluid-model solution is unique. In the Appendix we sketch a proof of (i). To establish (ii), it remains to be shown that any solution to the fluid-limit equation is linear. While we strongly conjecture that to be the case for any finite initial state Z (0), a rigorous proof appears challenging. Proof of Proposition 3.1. The proof of Proposition 3.1 follows from the next two lemmas.
Lemma 3.1. At time t, for any class i, a solution of Eq. (2) is given by z i (t) = m i t, where m i is a solution of Eq. (4). Proof. Suppose z i (t) = m i t, where m i is some constant. Substituting this into Eq. (2), we obtain that
983
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993 t R (m u) t i i du ds z i (t) = m i t = λi P Bi > mi u s 0 Z t Z t Ri (m) du ds = λi P Bi > mi u s 0 Z t s mi ds < log = λi P −Bi Ri (m) t 0 Z 1 mi mi −B −B P e i Ri (m) < u du = λi t 1 − E e i Ri (m) , = λi t
Z
Z
0
which yields that m i is a solution of Eq. (4).
Lemma 3.2. Eq. (3) has a unique solution m = (m 1 , . . . , m I ). Proof. We will establish uniqueness of R(m). The monotonicity of the LST’s βi (·) then implies uniqueness of m = (m 1 , . . . , m I ). A crucial role is played by a related optimization problem. To formulate this optimization problem, we rewrite the fixed-point equation (3) in the equivalent form Ri (m) mi = βi−1 , i = 1, . . . , I, (6) Ri (m) ρi ∗
where βi−1 (·) is the inverse of the Laplace–Stieltjes transform (LST) βi (y) = E[e−y Bi ], βi−1 (βi (y)) = y. Observe that the right-hand side only on m through Ri (m). This motivates us to introduce the function Hi (x), which depends α
has derivative Hi0 (x) = βi−1 ρxi . Since βi (x) is strictly decreasing in x, its inverse is strictly decreasing in x as well. Consequently, Hi (x) is strictly concave in x. Now, consider the optimization problem (Q)
H (R) =
maximize
I X
wi Hi (Ri )
i=1
subject to
A R ≤ C, R ≥ 0.
This optimization problem is strictly concave, and hence has a unique solution R = (R1 , . . . , R I ) (see for instance [7]). We proceed to show that the rate allocation vector R(m) = (R1 (m), . . . , R I (m)) satisfying Eq. (6) is a unique solution of the optimization problem (Q). We first consider the Karush–Kuhn–Tucker (KKT) necessary conditions [3] for problem (P). If Λ(z) is an optimal solution to problem (P), then there exist constants p j (z) ≥ 0 such that wi
zi Λi (z)
p j (z)
α
I X
=
J X
p j (z)A ji ,
if z i > 0,
(7)
j=1
! A ji Λi (z) − C j
= 0,
j = 1, . . . , J.
(8)
i=1
We now consider the KKT sufficient conditions for problem (Q). A feasible solution R ∗ = (R1∗ , . . . , R ∗I ) is a global optimum, if there exist constants p ∗j ≥ 0 such that wi
p ∗j
βi−1 I X i=1
Ri∗ ρi
α =
J X
p ∗j A ji ,
i = 1, . . . , I,
(9)
j=1
! A ji Ri∗
− Cj
= 0,
j = 1, . . . , J.
(10)
984
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
By definition, Ri (m) ≡ Λi (m) when m i > 0. It then follows from the KKT necessary conditions (7) and (8) for problem (P) that there exist constants p j (m) ≥ 0 such that wi
mi Ri (m)
α =
J X
p j (m)A ji ,
if m i > 0,
(11)
j=1
! p j (m)
X
A ji Ri (m) − C j
j = 1, . . . , J.
= 0,
(12)
i:m i >0
PI Now recall that Ri (m) ≡ ρi when m i = 0. Observing that i=1 A ji Ri (m) ≤ C j , j = 1, . . . , J , and writing PI P P A R (m) = A R (m) + A ρ , it may be inferred that if m i = 0 then for any j with i=1 ji i i:m i >0 ji i i:m i =0 ji i A ji = 1, we have the strict inequality X A ji Ri (m) < C j , (13) i:m i >0
and thus Eq. (12) forces p j (m) = 0. We deduce that J X
p j (m)A ji = 0,
if m i = 0,
(14)
j=1
and p j (m)
I X
! A ji Ri (m) − C j
= 0,
j = 1, . . . , J.
(15)
i=1
i = Rim(m) when m i > 0. Also, The fact that R(m) satisfies the fixed-point Eq. (6) implies that βi−1 Riρ(m) i = 0 when m i = 0. Substituting the latter equalities in Eqs. (11) and (14), respectively, we obtain βi−1 Riρ(m) i J Ri (m) α X wi βi−1 = p j (m)A ji , ρi j=1
i = 1, . . . , I.
(16)
Eqs. (15) and (16) show that R(m) satisfies the KKT sufficient conditions (9) and (10) for problem (Q), and hence is a global optimum. Besides proving uniqueness, the above equivalence also provides a way for actually computing the asymptotic growth rates m i : The rate allocation vector R(m) can be calculated by solving the concave programming problem (Q). There exist efficient interior-point methods for this, see for instance [4]. The asymptotic growth rates m i may then be determined from the rate allocations Ri (m) by a simple one-dimensional search. 3.2. Additional observations We now discuss some qualitative properties of the growth rates. (i) If the arrival rates and the flow sizes of all classes are scaled by common factors K > 0 and 1/K , respectively, thus keeping the traffic intensities constant, then the asymptotic growth rates scale by K . This makes sense as the scaling simply amounts to a change of timescale. (ii) Suppose we focus on a particular class and, dropping the class index, examine the impact of the variability of the flow size B on the asymptotic growth rate m for a fixed mean flow size E [B] and service rate R. It may be deduced from either form of the fixed-point equation that the growth rate is non-increasing in the variability of the flow size in the sense of the (LST) ordering. A random variable X is said to be larger or more variable than a random variable Y in the LST ordering if E[e−s X ] ≥ E[e−sY ] for all s ≥ 0. Note that the LST ordering is implied by the more common convex ordering, which provides a measure for the degree of variability of a distribution. This monotonicity property does not directly extend to a network setting where the service rate R depends on the growth rate m.
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
985
Fig. 1. Linear network.
(iii) If m i = 0, then the asymptotic growth rates m j , j 6= i, are identical to those in a corresponding system with both class-i traffic and all links j with A ji > 0 removed. The reason is that none of these links are bottlenecks if m i = 0. (iv) Suppose we consider a sequence of systems where the arrival rate of a particular class i in the kth system (k) is scaled in such a manner that limk→∞ λi = 0, while the flow size may be scaled in an arbitrary way, so that (k) it is not necessarily the case that limk→∞ ρi = 0. It may then be deduced from the fixed-point equation that (k) (k) limk→∞ m i = 0, and thus, because of (iii) above, m j → m j , j 6= i, with m j representing the asymptotic growth rate of class j in a corresponding system with class i removed. 4. Linear networks In this section we focus on the special case of a linear network as illustrated in Fig. 1. The network consists of links 1, . . . , L, each of unit capacity, and is offered traffic from classes 0, 1, . . . , L. Class- j flows require service from link j only, j = 1, . . . , L, while class-0 flows demand capacity on all links simultaneously. As mentioned earlier, linear networks provide a useful model for traffic that traverses several links and experiences bandwidth competition from independent cross-traffic. We assume that the load on at least one of the links exceeds the capacity, i.e., max j=1,...,L ρ j > 1−ρ0 . The bandwidth arbitration is governed by the proportional fair policy with unit class weights, PL i.e., the objective function G z (Λ) is given by i=0 z i log(Λi ). The capacity constraints take the form Λ0 + Λ j ≤ 1 for all j = 1, . . . , L. We are interested in determining the asymptotic growth rates m i of the various classes. According to Proposition 3.1, there exist non-negative coefficients (Lagrange multipliers) p j associated with the various links so that the m i and the corresponding rate allocations Ri together with the p j form a solution to the system of equations L X pj, m 0 = R 0 j=1 (17) m j = R j p j , j = 1, . . . , L , p j (R0 + R j − 1) = 0, j = 1, . . . , L , in conjunction with the set of fixed-point equations (6). The latter equations in fact allow us to express the m j in terms of the R j , yielding X L −1 R0 β = pj, 0 ρ0 j=1 (18) Rj β −1 = p , j = 1, . . . , L , j j ρj p j (R0 + R j − 1) = 0, j = 1, . . . , L . In total the above system provides 2L + 1 equations for 2L + 1 unknown variables Ri , i = 0, . . . , L, and p j , j = 1, . . . , L. In order to solve the above system of equations, we consider the nonempty subset J+ := { j : p j > 0} of links with strictly positive Lagrange multipliers. (The subset J+ cannot be empty, since that would imply ρ0 + ρ j ≤ 1 for all j = 1, . . . , J , and contradict the overload assumption.) Observe that p j = 0 means m j = 0. As observed in Property (iii) in Section 3.2, the growth rates of classes 0 and j ∈ J+ thus correspond to those in a scenario with classes j 6∈ J+ as well as links j 6∈ J+ removed. In particular, when J+ = { j+ }, the growth rates of classes 0 and j+ are identical to those in a single-node processor-sharing system with classes 0 and j+ .
986
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
For compactness, denote n = m0 ; n mj pj = , Rj
P
j∈J+
R j ≡ SJ+ = 1 −
R0 =
p j . Then the solution to the system of Eq. (17) may be represented as
m0 n
Rj = ρj
if j ∈ J+ ;
j = 1, . . . , J.
if j 6∈ J+ ; (19)
P Summing the last equality in Eq. (19) over j ∈ J+ , it follows that n = m 0 + j∈J+ m j . What remains is to determine the subset J+ in terms of the system parameters. Note that j ∈ J+ implies R0 + R j = 1, and thus necessitates ρ0 + ρ j ≥ 1. However, the latter inequality is not sufficient for j ∈ J+ , since it is possible that m j = 0 when other classes at other links sufficiently throttle the service rate of class 0. In order to characterize the subset J+ , observe that ρ j ≤ SJ+ for all j 6∈ J+ and ρ j > SJ+ for all j ∈ J+ . In view of the inherent symmetry, we may assume without loss of generality that the links are indexed such that ρ1 ≤ ρ2 ≤ · · · ≤ ρ L . Denote by σ j the common service rate obtained by classes j, . . . , L in a system with links j, . . . , L and classes 0 and m P L0 . Then the subset J+ is of the form { j+ , . . . , L}, with j, . . . , L only, σ j = 1 − Λ0 (m j , . . . , m L ) = 1 − m0+
k= j
mk
j+ := max{ j : ρ j−1 ≤ σ j }. In the case B0 ≡ B L , it is easily verified that σ L = λ L /(λ0 + λ L ). The above characterization of the subset J+ may be interpreted as follows. If ρ j−1 ≤ σ j , then competition from classes j, . . . , L alone against class 0 is sufficient to throttle the rate of class 0 to an extent that what remains available for classes 1, . . . , j − 1 exceeds their respective loads, and hence m 1 = · · · = m j−1 = 0. This scenario occurs when the loads of classes 1, . . . , j − 1 are relatively low and the loads of classes j, . . . , L are sufficiently high. Note that this may occur even when ρ0 + ρi > 1 for some classes i = 1, . . . , j − 1. Although these classes rely on service from overloaded links, they remain stable thanks to the much stronger competition at other higher-loaded links. In contrast, if ρ j−1 > σ j , then competition from classes j, . . . , L alone is not sufficient to provide stability to class j − 1, and hence m j−1 > 0. 4.1. Exponential flow sizes In order to determine the growth rates explicitly, we need to specify the flow size distributions of the various classes that occur in the set of fixed-point equation (6). In the case of exponential flow sizes, substituting (6) into (19), the growth rates of the various classes may be represented in terms of the single variable n as m 0 = λ0 − µ0
m0 λ0 n , = n µ0 + n
m j = λj − µj
λ0 n − m0 = λj − µj 1 − n µ0 + n
(20)
,
j ∈ J+ .
(21)
Summing the above equations results in X λ0 n λ0 n= + λj − µj 1 − , µ0 + n µ0 + n j∈J +
which yields the quadratic equation n 2 + νn + κ = 0,
(22)
with ν := µ0 +
X j∈J+
κ :=
X j∈J+
µ j − λ0 −
X
λj,
j∈J+
µ j (µ0 − λ0 ) − µ0
X
λj.
j∈J+
Substituting the positive solution in Eqs. (20) and (21) gives expressions for the asymptotic growth rates. To see that there is indeed a unique positive solution, recall that a quadratic equation has a unique positive solution when the
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
987
Fig. 2. Star network with three links.
zero-order constant is nonpositive, which may be written as X µj ρ0 + ≥ 1. ρj P µj j∈J +
Noting that X j∈J+
µj ρj P j∈J+
(23)
j∈J+
µj
≥
X j∈J+
µj min ρk P
k∈J+
µj
= min ρk , k∈J+
j∈J+
the inequality (23) is seen to hold by virtue of the fact that ρ0 + ρ j ≥ 1, j ∈ J+ . 5. Star networks We now turn attention to the special case of a star network as depicted in Fig. 2. As mentioned earlier, star networks offer a convenient abstraction for scenarios where the core is highly over-provisioned and congestion predominantly occurs at the edge with comparatively low-capacity access links. The network is composed of L links, each of unit capacity, and is offered traffic from L(L − 1)/2 classes labeled as {i, j}, i, j = 1, . . . , L, i 6= j. The route of class {i, j} simply consists of the two links i and P j. We assume that the load on at least one of the links exceeds the capacity, i.e., max j=1,...,L σ j > 1, with σ j := k6= j ρ{ j,k} . The bandwidth arbitration P is governed by the proportional fair policy with unit class P weights, i.e., the objective function G z (Λ) is given by j6=k z { j,k} log(Λ{ j,k} ). The capacity constraints take the form k6= j Λ{ j,k} ≤ 1 for all j = 1, . . . , L. For star networks, Proposition 3.1 and Eq. (6) imply that the Lagrange multipliers p j associated with the links in the network and the corresponding rate allocations R{ j,k} satisfy the following system of equations R{ j,k} −1 β{ j,k} ρ{ j,k} = R{ j,k} ( p j + pk ), j 6= k, ! (24) X p R − 1 = 0, j = 1, . . . , L . j { j,k} k6= j
In total the above system provides L(L + 1)/2 equations for L(L + 1)/2 unknown variables R{ j,k} , j 6= k, and p j , j = 1, . . . , L. In the case of exponential flow sizes, the set of fixed-point equations takes the explicit form in Eq. (5). The above system of equations then simplifies to λ{ j,k} − µ{ j,k} R{ j,k}! = R{ j,k} ( p j + pk ), j 6= k, X (25) R{ j,k} − 1 = 0, j = 1, . . . , L . pj k6= j
As before, we need to consider the subset of links with strictly positive Lagrange multipliers in order to solve the above system of equations.
988
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
5.1. Three-link network As an illustrative example, we now focus on the case of a triangular network with three links and three classes as depicted in Fig. 2. In that case we need to distinguish three scenarios, (I), (II) and (III), depending on whether one, two or all three of the Lagrange multipliers P are strictly positive, respectively. It cannot occur that all three Lagrange multipliers are zero, since that would imply k6= j ρ{ j,k} < 1, j = 1, 2, 3, and contradict the overload assumption. With minor abuse of notation, we define m i := m {1,2,3}\{i} , Ri := R{1,2,3}\{i} , and ρi := ρ{1,2,3}\{i} . The above system of equations in (24) may then be rewritten as m i = Ri ( p j + pk ), {i, j, k} = {1, 2, 3}, (26) p j (Ri + Rk − 1) = 0, {i, j, k} = {1, 2, 3}. For compactness, denote m := m 1 + m 2 + m 3 , and n = m i + m j . The solutions in the above three scenarios may be then represented as mi m j , , ρk , (I) Ri , R j , Rk = n n pi = p j = 0, pk = n, mi m j + mk m j + mk (II) Ri , R j , Rk = , , , m m m mj mk pi = 0, pj = m, pk = m, m j + mk m j + mk 1 (III) R1 = R2 = R3 = , 2 X pi = m j − m i > 0, i = 1, 2, 3. j6=i
The above results reveal an interesting trichotomy in the behavior of the triangular network. In case (I) the network behaves as a single-node processor-sharing system with classes i and j only. The conditions for case (I) to occur in terms of the system parameters also coincide with the corresponding ones in the linear network with |J+ | = 1. Case (II) corresponds to the case of the linear network with |J+ | = 2. The conditions for this case to arise subsume the corresponding ones in the linear network, but include an additional condition that the loads of the three classes should be slightly unbalanced. If the latter condition is violated, i.e., the loads of the three classes are nearly equal, then case (III) arises, which has no counterpart in the linear network. In this case, each of the three classes behaves as in an isolated processor-sharing system with capacity 21 . 6. Numerical results In this section we discuss the numerical experiments that we performed to corroborate and illustrate the analytical findings. We present simulation results to demonstrate the convergence of the scaled number of flows to the fluid limit. In addition, we examine the impact of the traffic intensities and the variability of the flow sizes on the asymptotic growth rates. Throughout the numerical experiments we focus the attention on a two-link linear network operating under the proportional fair policy with unit class weights. 6.1. Exponential flow sizes We first consider the case of exponential flow sizes, and specifically investigate the impact of the traffic intensity on the asymptotic growth rates. Figs. 3 and 4 plot the asymptotic growth rates m 0 , m 1 , m 2 for exponential flow sizes as a function of the traffic intensity. We let all classes have the same mean flow size and let the arrival rates vary. Fig. 3 shows the growth rates in a situation where the loads of classes 0 and 2 are fixed and the load of class 1 is varied, and reveals natural qualitative trends. As the load of class 1 increases, the competition with class 0 becomes stronger, and as a result both queues
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
989
Fig. 3. Asymptotic growth rates m 0 , m 1 , m 2 as a function of ρ1 .
Fig. 4. Asymptotic growth rates m 0 , m 1 , m 2 as a function of ρ0 .
grow at a higher rate. The reduced service rate of class 0 in turn leaves more capacity available for class 2, and thus its growth rate decreases, ultimately reaching stability. Fig. 4 shows the growth rates in a situation where the loads of classes 1 and 2 are fixed and the load of class 0 is varied. As the load of class 0 increases, both classes 1 and 2 receive less service. Consequently, all three classes build up queues at a higher rate. In particular, the figures illustrate the stability properties of the linear networks discussed in Section 4. Recall that the link overload conditions ρ0 + ρ1 > 1, ρ0 + ρ2 > 1, are necessary but not sufficient for the number of flows z 1 or z 2 to grow. For instance, in Fig. 3(b) the asymptotic growth m 1 becomes positive only when ρ1 reaches a value of 0.3. While the first link is already overloaded due to the large number of class-0 flows, the strong competition with class 2 provides sufficient bandwidth for class 1 to remain stable. Note that if m 1 = 0 the network behaves as a single-node processor-sharing system with classes 0 and 2 only. Thus, the stability condition for class 1 in this case is determined 2 as ρ1 < 1 − Λ0 = ρ0ρ+ρ . 2 Fig. 5 plots the value of Z i (t)/t as a function of t obtained by simulation. The horizontal lines represent the asymptotic growth rates m 0 , m 1 , and m 2 as computed from Eq. (22). All classes have exponential flow sizes with unit mean. The traffic intensities are ρ0 = 1.2, ρ1 = 0.5, ρ2 = 0.7.
990
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
Fig. 5. Scaled queue length
Z i (t) t as a function of time, ρ0 = 1.2, ρ1 = 0.5, ρ2 = 0.7.
6.2. Hyperexponential flow sizes We now turn to the case of hyperexponential flow sizes, and investigate the impact of the variability of the flow sizes on the asymptotic growth rates by varying the parameter values of the hyperexponential distribution. The flow sizes of all classes have the same hyperexponential distribution: the flow size is exponential with mean 1/ν1 with probability p, and exponential with mean 1/ν2 otherwise. Moreover, we assume that the contributions to the mean are balanced, i.e., p/ν1 = (1 − p)/ν2 , so that as p → 0, ν1 = E2 pB → 0 and ν2 → E2B . We fix the mean flow size E [B] = 2, and vary the value of p in the interval [0, 1/2]. Note that when p = 21 , the flow size becomes simply an exponential random variable with mean 2 and squared coefficient of variation σ 2 = 1. However, as p tends to 0, the squared coefficient of variation grows like 1/ p. Fig. 6 plots the growth rates m 0 , m 1 , m 2 as functions of the squared coefficient of variation. Two limiting cases are shown as the markers on the graphs which provide lower and upper bounds for the asymptotic growth rates; white markers show the growth rates computed for exponential flow sizes with mean 2 (the case p = 21 ); the black markers indicate the growth rates obtained for exponential flow sizes with mean 1. The observation that the growth rates approach those for exponential flow sizes with mean 1 may be explained as follows. Note that a particular class i with arrival rate λi and a hyperexponential flow size distribution with parameters µi1 , µi2 , pi1 and pi2 , with pi1 + pi2 = 1 may be equivalently replaced by two classes with arrival rates λik = λi pik and exponential flow sizes with parameter µik , k = 1, 2. Now suppose that we consider a regime where the coefficient of variation grows large by letting pi1 , µi1 ↓ 0, with pi1 /µi1 = ci1 and pi2 /µi2 = ci2 fixed, meaning that µi2 ↑ 1/ci2 . It then follows from the observation described in (iv) in Section 3 that the growth rates of the three original classes with hyperexponential flow sizes, in the limit are identical to those in a scenario with three classes with arrival rates λi2 and exponential flow sizes with parameter 1/ci2 . In the numerical experiments, we have focused the attention on an admittedly simple two-link linear network. Nevertheless, the network model already appears to be sufficiently rich to reveal several interesting features and similar qualitative properties as may be expected to occur in more complex scenarios. 7. User impatience Let us now consider an extension of the model to a scenario with user impatience. Impatient users may abandon the system before completing service. An impatient user is characterized by a positive random initial lead time in addition to its flow size. Denote by Di the initial lead time of class-i flows. An impatient user has a deadline (arrival time plus
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
991
Fig. 6. Asymptotic growth rates m 0 , m 1 , m 2 as functions of the squared coefficient of variation.
initial lead time); the user leaves the system either when it completes service or when the deadline expires, whichever occurs first. As mentioned earlier, user impatience has a particularly pronounced impact in overload conditions. The single-link single-class version of this model has been studied in [6,11,12]. Following similar arguments as in those papers, we propose to approximate the number of flows of each class by the solution of the fixed-point equation zi Bi . (27) z i = λi E min Di , Ri (z) This equation may be heuristically explained as follows. Let Z ir be the steady-state number of class-i flows in the r th system, and let Vir (B) be the sojourn time of a user that does not abandon. Assume that users are relatively patient, i.e., let their initial lead time also be scaled as Di r . Then the actual sojourn time is given by min{Vir (B), Di r }, and Little’s law implies E[Z ir ] = λE[min(Vir (B), Di r )].
(28)
Divide both sides of the equality by r . Since we observe the system in steady state at time 0, the number of flows hardly changes over the course of a sojourn time, and by the so-called ‘snapshot principle’ we conclude that Zr Vir = Ri (Zi r ) B + o(r ). Noting that Z ir /r → z i then gives Eq. (27) after dividing both sides of (28) by r and letting r → ∞. To make these heuristics rigorous, would require us to unify the frameworks of [11,13]. In addition, it is not clear a priori that the fixed-point Eq. (27) has a unique solution in general. This is a subject of current research. At this point, we focus on a simple example. It was shown in [6,11,12] that for a single link, impatience can have a substantial impact, especially if the initial lead time is a constant times the flow size. In that case, the abandonment rate is one hundred percent. To see whether this holds in network scenarios as well, we assume that Di = θ Bi , with some constant θ > 0, and consider the case of a two-link linear network operating under the proportional fair policy with unit class weights. Then the equations simplify to z 0 = ρ0 min{θ, z 0 + z 1 + z 2 }, zi z i = ρi min θ, (z 0 + z 1 + z 2 ) , z1 + z2
i = 1, 2.
(29)
In addition, the constraints R0 + Ri ≤ 1 should hold for a solution of the fixed-point equation (29) to be admissible. If ρ2 < ρ1 /(ρ0 + ρ1 ) and ρ0 + ρ1 > 1, then it follows that z 2 = 0, z 0 = θρ0 , z 1 = θρ1 is a feasible solution to the fixed-point equation (29). We therefore conclude that the overall abandonment rate can be lower than one hundred percent in network scenarios, due to the fact that some classes may only traverse links with surplus capacity.
992
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
Appendix We now sketch a proof of the fact that any convergent subsequence of (Z (r t)/r, t ≥ 0) is a fluid-model solution in our sense. The complete proof details will be provided in an extended version of the paper. Assume that r → ∞ through a specific sequence. We need to show that the limit is a fluid-model solution. Define Z¯ ir (t) = r1 Z i (r t). From Eq. (5.2) on page 18 of [13], we obtain, r E i (t) 1 X Z¯ ir (t) = F0r (t) + I (Bi,k > S¯ir (Uik /r, t)), r k=1 ¯r
(30)
r , k ≥ 1) where (E ir (t), t ≥ 0) denotes the number of class-i flows that arrive during the time interval (0, t], and (Uik r is the arrival epoch of the kth class-i flow. Function F0 (t) represents the number of customers active at time 0 that are still in the system at time t. The left-hand side converges to, say, z i (t). The first term on the right-hand side is bounded from below by 0 and from above by Z i (0)/r . Consequently, its limit as r → ∞ is 0. To deal with the second term on the right-hand side, we use a similar approach as in [11]: The time interval [0, t] is partitioned into small slices of length 1/N (for some large constant N ). For each slice, an asymptotic lower and upper bound is derived, using the law of large numbers for the empirical distribution of the flow sizes. By letting N → ∞, the limits are shown to coincide.
References [1] E. Altman, T. Jimenez, D. Kofman, DPS queues with stationary ergodic service times and the performance of TCP in overload, in: Proc. IEEE Infocom 2004, Hong-Kong, 2004. [2] V. Anantharam, How large delays build up in a GI/G/1 queue, Queueing Syst. 5 (1989) 345–367. [3] M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley, New Jersey, 1993. [4] A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization, in: MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2001. [5] T. Bonald, L. Massouli´e, Impact of fairness on internet performance, in: Proc. ACM Sigmetrics & Performance 2001 Conf., Boston, MA, USA, 2001, pp. 82–91. [6] T. Bonald, J.W. Roberts, Performance modeling of elastic traffic in overload. in: Proc. ACM Sigmetrics & Performance 2001 Conf., Boston, MA, USA, 2001, pp. 342–343. [7] S. Boyd, L. Vanderberghe, Convex Optimization, University Press, Cambridge, 2004. [8] M. Bramson, Instability of FIFO queueing networks, Ann. Appl. Probab. 4 (1994) 414–431. [9] M. Bramson, Stability of networks for max–min fair routing, in: Presentation at the 13th INFORMS Applied Probability Conference, Ottawa, 2005. [10] G. Fayolle, A. De la Fortelle, J.-M. Lasgouttes, L. Massouli´e, J.W. Roberts, Best-effort networks: Modeling and performance analysis via large network asymptotics, in: Proc. IEEE Infocom 2001, 2001. [11] H.C. Gromoll, Ph. Robert, A.P. Zwart, Fluid limits for processor sharing queues with impatience, 2006 (submitted for publication). [12] H.C. Gromoll, Ph. Robert, A.P. Zwart, R. Bakker, The impact of reneging in processor sharing queues, in: Proc. ACM Sigmetrics & Performance 2006 Conf., St. Malo, France, 2006. [13] H.C. Gromoll, R.J. Williams, Fluid limit of a network with fair bandwidth sharing and general document size distribution, Ann. Appl. Probab. (2007) (in press). [14] H.C. Gromoll, R.J. Williams, Fluid model for a data network with alpha-fair bandwidth sharing and general document size distributions: Two examples of stability, 2007, Preprint. Accepted for IMS Festschrift volume in honor of Tom Kurtz. [15] A. Jean-Marie, Presentation at the 30th Conference on the Mathematics of Operations Research, Lunteren, The Netherlands. http://www.lirmm.fr/ ajm/Talks/18Jan2005/transparents.pdf, 2005. [16] A. Jean-Marie, Ph. Robert, On the transient behavior of the processor sharing queue, Queueing Syst. 17 (1994) 129–136. [17] F.P. Kelly, R.J. Williams, Fluid model for a network operating under a fair bandwidth-sharing policy, Ann. Appl. Probab. 14 (2004) 1055–1083. [18] S.H. Lu, P.R. Kumar, Distributed scheduling based on due dates and buffer priorities, IEEE Trans. Automat. Control 36 (1991) 1406–1416. [19] L. Massouli´e, Structural properties of proportional fairness: Stability and insensitivity, 2005, Preprint. [20] L. Massouli´e, J.W. Roberts, Bandwidth sharing: Objectives & algorithms, in: Proc. IEEE Infocom ’99, New York, NY, USA, 1999, pp. 1395–1403. [21] J. Mo, J. Walrand, Fair end-to-end window-based congestion control, IEEE/ACM Trans. Netw. 8 (2000) 556–567. [22] A.L. Puha, A.L. Stolyar, R.J. Williams, The fluid limit of an overloaded processor sharing queue, Math. Oper. Res. 31 (2006) 316–350. [23] J.W. Roberts, L. Massouli´e, Bandwidth sharing and admission control for elastic traffic, in: Proc. ITC Specialist Seminar, Yokohama, Japan, 1998. [24] G. De Veciana, T.-L. Lee, T. Konstantopoulos, Stability and performance analysis of networks supporting services with rate control — could the Internet be unstable? in: Proc. IEEE Infocom ’99, New York, NY, USA, 1999, pp. 802–810.
R. Egorova et al. / Performance Evaluation 64 (2007) 978–993
993
[25] G. De Veciana, T.-L. Lee, T. Konstantopoulos, Stability and performance analysis of networks supporting elastic services, IEEE/ACM Trans. Netw. 9 (2001) 2–14. Regina Egorova received her Bachelor degree from the Department of Applied Mathematics, Ufa State Aviation Technical University, Russia, in 2002. In 2004 she received her M.Sc. degree in Risk Analysis and Environmental Modelling from Delft University of Technology, The Netherlands. Currently, she is a Ph.D. candidate employed by CWI and Eindhoven University of Technology. Her research interests are in the analysis of resource sharing in computer and communication networks.
Sem Borst received the M.Sc. degree in applied mathematics from the University of Twente, The Netherlands, in 1990, and the Ph.D. degree from the University of Tilburg, The Netherlands, in 1994. During the fall of 1994, he was a visiting scholar at the Statistical Laboratory of the University of Cambridge, England. In 1995, he joined the Mathematics of Networks and Systems department of Bell Laboratories, Lucent Technologies in Murray Hill, USA. He also holds a part-time appointment as a professor of Stochastic Operations Research at Eindhoven University of Technology. Sem Borst is a member of IFIP Working Group 7.3, and serves or has served as a member of several program committees and editorial boards. His main research interests are in the area of performance evaluation and resource allocation algorithms for communication networks.
Bert Zwart is Coca Cola Associate Professor of Industrial and Systems Engineering at Georgia Tech. Bert holds an M.A. in Econometrics from the Free University, Amsterdam, and a Ph.D. in Applied Mathematics from Eindhoven University of Technology. His research is concerned with modeling, analysis and simulation of stochastic systems arising in actuarial and financial mathematics, computer and communication systems, manufacturing systems, and service systems. He serves on the editorial boards of Operations Research, Mathematics of Operations Research, and Mathematical Methods of Operations Research.
