Integrating Streaming and File-transfer Internet Traffic ... - CiteSeerX

Integrating Streaming and File-transfer Internet Traffic: Fluid and Diffusion Approximations Sunil Kumar∗and Laurent Massouli´e

†

November 22, 2005

Abstract We study the behavior of a communication link that divides its capacity fairly among file transfers and streaming flows. We identify appropriate asymptotic regimes and obtain fluid and diffusion limits. The non-trivial, yet tractable, limits can be used for performance analysis and policy design. Furthermore, the scalings used to obtain these limits are themselves of interest and help explain phenomena observed in previous simulation studies.

1

Introduction and Summary

Traffic carried over modern communication networks, such as the Internet, is conveniently separated into two classes. Streaming, or real-time traffic is characterized by data flows with an intrinsic duration, although the associated traffic volume may vary, if variable rate coding is used. In contrast, file-transfer, or data traffic is characterized by data flows with an intrinsic volume, but the associated flow duration may vary, depending on the rate at which transfer proceeds. An interesting question concerns the integration of these two traffic classes. Candidate integration policies, where streaming flows are allocated a fair share of the available capacity, are discussed for instance in [2, 3, 6]. In particular, in [6] the authors describe formal fluid models of a candidate fair traffic integration strategy. They also report simulation results of stochastic systems that support the following conjectured claim. Given a target amount of file-sharing traffic to be carried and a target mean response time for such file sharing flows, fluctuations in the system state are reduced by increasing the amount of streaming traffic integrated with the file sharing traffic. In this paper, we aim to gain an understanding of the effects of integration. To this end, we provide a complete asymptotic analysis of the fluid and diffusion limits for a single communication link of capacity C carrying a mixture of file-sharing and streaming flows, and sharing its capacity equally among all the flows. Streaming flows arrive at the link at rate κ and are “served” at rate η, that is, they hold for a mean time of η1 regardless of the capacity alloted to them. File transfers arrive at a rate ν and require µ1 amount of service before they depart from the link; the actual ∗ †

Graduate School of Business, Stanford University, [email protected] Microsoft Research, Cambridge, U.K., [email protected]

1

holding time depends on the capacity allocation. We study a Markovian model in which all arrival and service processes are independent Poisson processes. In this case, we can consider the capacity allocation policy to be head-of-line for purposes of analysis, and assume that all the capacity alloted to the file transfers is given to the file at the head of the line. We are interested in the large system asymptotics of this system, when C, κ and ν increase without bound. If we were to consider the streaming flows in isolation, then the system would be described by a M/M/∞ queue, while the M/M/1 processor sharing queue captures the performance of file transfer traffic in isolation, if we assume available capacity is shared fairly between competing users (see [9] for example). As is well known, for file sharing traffic in isolation, in the heavy traffic asymptotic ν approaches one, the natural diffusion approximation is reflected Brownian regime where ρ = µC motion. In contrast, for streaming traffic on its own, the natural diffusion approximation in the asymptotic regime where κ → ∞ is an Ornstein-Uhlenbeck process. For the case when the number of streaming flows is fixed at some non-zero level, previous results [11] show that the diffusion limit for file transfers in heavy traffic is a Bessel process. In our model, where both types of flows vary stochastically, the limits obtained lie somewhere in between these extreme cases, as expected. We obtain fluid and diffusion limits that are valid regardless of whether the system is in heavy traffic or not. In particular, the fluid limit obtained in heavy traffic resembles the drift function of a Bessel process. The diffusion limit obtained is a coupled two-dimensional Ornstein-Uhlenbeck process. This process is Gaussian if the initial states are Gaussian and its stationary distribution can be easily obtained. The key technical difficulty that precludes the direct adoption of previously developed proof techniques, such as [8], in obtaining our asymptotic results is that the server effort allocation function is not Lipschitz in the state. While obtaining fluid and diffusion limits are the primary goals of our analysis, the scaling and centering used themselves provide a lot of information about the system. First, we identify κ ¯ ≈ ν that the mean number of file transfers in progress N η(µC−ν) , and thus by Little’s law, the κ . This implies that mean number of file transfers mean response time for file transfers R ≈ η(µC−ν) is the mean number of streaming flows multiplied by the mean number of file transfers in the absence of any streaming flows. We also identify that there is a time scale separation between the streaming flows and the file transfers, with a unit of time on the file transfer time scale being equal κ to τ = µC(1−ρ) 2 time units on the streaming time scale. The magnitude of the fluctuations of the √ q κ ρ ρ2 file transfer process are captured by σN ≈ (1−ρ) η(τ +1) + η . To further illustrate the utility of the scaling and centerings themselves, consider the following “stabilization effect” investigated using simulation in [6]. The offered file transfer traffic, i.e. ν, µ, is fixed, as is the target mean response time R for such file transfers. The parameter η characterizing the duration of individual streaming connections is also fixed. The amount of streaming traffic, captured by κ, is allowed to vary, as is the capacity of ithe link to ensure that R remains fixed. The system capacity C varies with κ as h 2 κ 1 C = µ ηR + ν . Then, the time scale of the file transfers is given by τ = ηR + ν(ηR) κ . Thus as κ becomes small relative to ν, that is, the volume of file transfers increasingly dominates that of the streaming flows even though both increase without bound, the time scale for the file transfers becomes increasingly large. Moreover, for small κ relative to ν, the magnitude of the fluctuations √ of the file transfers becomes σN ∼ νR η √1κ . Thus, when the streaming traffic is reduced keeping 2

the target response time constant, the file transfers display increasingly large, slow fluctuations, as observed in Figure 1 of [6], validating the stabilizing effect of the streaming flows. We note that the asymptotic analysis carried out in this paper applies, essentially without modification, to weighted allocation rules that share capacity between streaming flows and file transfer by weighting each streaming flow as if it were θ flows in the fair sharing rule, for a design parameter θ > 0. This allows us to evaluate the impact of θ on system performance, along the lines of [3]. Finally, the assumption that the streaming flows join regardless of the bandwidth alloted to them is stylized. Streaming flows may require a minimum allocation. Our asymptotic analysis does allow us to determine when this assumption is moot. The fluid limits tell us that the allocation to in equilibrium. If this ratio grows without bound or converges to a the streaming flows is C(1−ρ) κ value greater than the minimum allocation then our assumption is reasonable since the minimum allocation constraint is not binding for the majority of the flows. Otherwise, one would need to take this constraint into account in the analysis. The paper is organized as follows. Section 2 provides some motivation for the main results by discussing a known result due to Yamada for fixed number of streaming flows; Section 3 describes the system model; Section 4 contains the main results, with the identification of fluid and diffusion approximations, and of limiting distributions. Proofs are given in Sections 5 and 6.

2

Motivation: the fixed M case

Before we consider the general system model, the following simple case helps illustrate the nature of the system that we study in the paper. Consider a system where file transfers arrive at a single link according to a Poisson process at rate ν. Each file transfer requires an independent and identically distributed service time with mean µ1 . The service times of the file transfers are independent of the arrival process. Furthermore there are always exactly M streaming or persistent flows that use the same link, where M is some non-negative integer. The link has capacity C that is equally shared among the file transfers and the streaming flows. Under the assumption that all service time distributions are i.i.d exponential, the distinction between whether the link splits its capacity equally among all transfers present, or simply gives it to the file transfer at the head of the line is moot. The reader is best served by thinking of a head-of-line allocation. If N (t) denotes the number of file transfers at time t, then N (.) is a Markov chain on {0, 1, 2, . . . , }. Its rate matrix Q i+1 for i = 0, 1, 2, . . . and Q(i, j) = 0 otherwise. is given by Q(i, i + 1) = ν and Q(i + 1, i) = µC i+1+M We assume that ν < µC. Now consider a sequence of such systems indexed by a (dummy) integer parameter L, each with parameters ν L and C L > ν L , keeping µ fixed. Consider the asymptotic parametric regime where ν L → ν and C L → C as L → √ ∞.L Furthermore consider the heavy traffic asymptotic regime L L L where v = µC − ν → 0, and Lv → d for some d ∈ R, as L → ∞. Let W (t) denote the amount of work in system. That is, W L (t) denotes the amount of time required to complete all the file transfers currently in the system. Then the following result follows directly from Theorem 1 of [11]. (Here, and elsewhere in the paper, ⇒ denotes weak convergence in the topology of uniform convergence on compact time sets.)

3

L

L

(0) (L·) Proposition 1 (Yamada). If W√2L ⇒ w(0), then W√2L ⇒ w(·) where w(·) is a diffusion whose 2 square process, z(·) = w (·), satisfies Z t Z tp  √ p z(s)dB(s), (1) 1 + M − 2d z(s) ds + 2 z(t) = z(0) + 0

0

where B(·) is a standard Brownian motion.

When d = 0 the process w is called a Bessel process. Furthermore, when M = 0, w is a reflected Brownian motion. The process w has the infinitesimal generator   d M 1 −√ T g(x) = g′ (x) + g′′ (x), 2x 2 2 for all twice continuously differentiable functions on R with compact support g that can be expressed as g(x) = g˜(x2 ) for some twice continuously differentiable function g˜ on R with compact support. Yamada’s result provides an excellent illustration of the nature of the results that we will obtain in the sequel. First, in the absence of persistent streams, i.e. when M = 0, the behavior of the system is identical to that of an M/M/1 queue. However, the addition of a constant M > √ 0 causes a substantial change in system behavior on an increasingly large spatial scale of order L, when the system approaches heavy traffic. This is akin to the behavior observed in polling models [1]. Moreover the limit identified in Proposition 1 is quite amenable to analysis [11]. For example, when d > 0, the process in (1) is positive recurrent, and has a stationary distribution with gamma density given by [1] √ √ √ 2|d|( 2|d|)M e− 2|d|x , x ≥ 0. π(x) = M! Given that the workload for an M/M/1 queue is exponentially distributed in steady-state, we see that the workload in the link with M persistent streams behaves an M -fold convolution of the workload without persistent streams in steady-state in heavy traffic1 . This fact will prove useful in identifying the correct spatial scaling for fluid limit and centering for the diffusion limit in the sequel. In the following sections we will no longer constrain the streams to be fixed at M . Rather we will allow them to arrive and depart at random, and carry out analysis similar to the one described in this section. We will use this analysis to quantify the impact of streaming flows on system behavior.

3

System Model

As in the previous section, we consider a system where file transfers arrive at a single link according to a Poisson process at rate ν. Each file transfer requires an independent and identically distributed service time with mean µ1 . The service times of the file transfers are independent of the arrival process. Unlike the previous section, the number of streaming flows is not fixed. Rather streaming 1

This fact could be obtained by analyzing the birth-death processes in the original system without resorting to limiting analysis in heavy traffic.

4

flows arrive at the link according to a Poisson process of rate κ, independent of everything else in the system. Each streaming flow persists for an independent and identically distributed amount of time with mean η1 , independent of everything else in the system. The link has capacity C that is equally shared among the file transfers and the streaming flows. While the time taken to complete a file transfer is dependent on the capacity allocated, the time for which a streaming flow persists in the system does not depend on the capacity allocated. This is the model proposed in [6]. As before, under the assumption that all service time distributions are i.i.d exponential, the distinction between whether the link splits its capacity equally among all transfers present, or simply gives it to the file transfer at the head of the line is moot. Let (M (t), N (t)) ≥ 0 denote the number of streaming flows and file transfers respectively present in the link at time t ≥ 0. Under our assumptions, (M (t), N (t)) is a Markov chain on Z2+ with rate matrix  κ if m′ = m + 1, n′ = n    ′  n′ = n
 mη if m′ = m − 1, ′ ′ ′ ν if m = m, n = n + 1 (2) Q (m, n), (m , n ) =  Cµn ′ = m, n′ = n − 1, n > 0   if m   m+n 0 otherwise. The following alternate representation of the system under consideration will prove useful in some of the analysis.   Z t M (t) = M (0) + Ξ3 (κt) − Ξ4 η M (s)ds , and 0   Z t N (s) ds , (3) N (t) = N (0) + Ξ1 (νt) − Ξ2 µC 0 N (s) + M (s)

where Ξi , i = 1, 2, 3, 4 are independent unit rate Poisson processes. The convention 0/0 = 0 is used to specify the integrand in this last equation when N (s) = M (s) = 0. In our analysis we will consider a centering and scaling scheme of the form  

¯ , 1 N (τ t) − N ¯ ˆ (t), N ˆ (t)) = 1 M (τ t) − M (4) (M sm sn

¯, N ¯ ≥ 0 and sm , sn , and τ > 0. This transformation preserves the Markov for some choice of M ˆ (·), Nˆ (·)) being a Markov chain with the obvious transformation of the rate matrix structure with (M in (2). Finally, note that the number of streaming flows M is simply the contents of a M/M/∞ queue. The process M (·) is not influenced by the process N (·). In this sense, the dynamics of N (·) are modulated by the modulating process M (·).

4

Asymptotic Analysis

In order to carry out the asymptotic analysis of the system specified in (2) we will consider a sequence of independent systems indexed by a dummy integer parameter L. In the L-th system, we assume that the parameter 5-tuple (κ, η, ν, µ, C) is given by (κL , η, ν L , µ, C L ). That is, we keep 5

µ and η fixed and let κ, ν, C vary with L. We make the following assumptions on the sequence of parameters. (A1) µC L > ν L for each L. That is, we assume that each of the systems has a nominal load due νL to file transfers, ρL := µC L that is strictly smaller than 1. This is a necessary condition for stability of the system. (A2) κL → ∞, ν L → ∞, and C L → ∞ as L → ∞. That is, we look at systems that handle increasingly faster arrivals, and have increasing larger capacity. Our asymptotic analysis, therefore, can be considered “fast system” analysis. (A3) ρL → ρ ≤ 1. That is, we do not preclude the systems from approaching heavy traffic. (A4) τ L :=

κL µC L (1−ρL )2

→ τ ∈ [0, ∞].

As long as (A1)-(A4) are satisfied, we do not need to distinguish between the asymptotic regimes under normal load, when ρ < 1 or heavy traffic, when ρ = 1. The asymptotic results are summarized in the two propositions below. Proposition 2 (Fluid Limits). 1) Assume that κL → ∞, and M L (0)/κL → m(0) almost surely (resp. in probability) as L → ∞. Then almost surely (resp. in probability) we have the convergence mL (·) :=

1 M L (·) → m(·) κL

uniformly on compact time sets

(5)

where m(·) is the unique solution of the equation m(t) = m(0) + t − η

Z

t

m(s)ds, 0

t ≥ 0.

(6)

2) Assume that (A1-A4) hold. Consider the time and space rescaled process nL (·) =

1 L L N (τ ·), sL n

L

κ L L where τ L is as in (A4) and sL n = 1−ρL . If (m (0), n (0)) converges in probability as L → ∞ to some deterministic (m(0), n(0)) both of which are strictly positive, then in probability

nL (·) → n(·)

uniformly on compact time sets

where n(·) is the unique solution of the equation Z t ρm(τ s) − n(s) ds, n(t) = n(0) + 0 (1 − ρ)m(τ s) + n(s)

t ≥ 0.

(7)

(8)

In this last equation, m(·) denotes the solution of (6). When τ = +∞, m(sτ ) is understood to be m(∞) = 1/η, that is the limit of m(s) as s → ∞. 6

Note that the first part of the proposition concerning the M -component is a well-known result for M/M/∞ queues. It follows for instance from Theorem 8.1, p. 52 of Kurtz [7] on so-called density dependent jump Markov processes, the assumptions of which are trivially satisfied in the present case. The proof of the second part of Proposition 2 is postponed to Section 5. Some comments on Proposition 2 are in order. First, consider (6-8). Then (6-8) has a unique solution for each (m(0), n(0)) > 0 which converges to the unique equilibrium point ( η1 , ηρ ). One can argue along the lines of [6] that this equilibrium point is globally attractive in R2+ . Second,(6-8) displays a time scale separation between the m and n, with the m-component operating on a time scale that is τ times faster than the n component. When τ = +∞, the m component is infinitely faster, achieving equilibrium instantaneously. Third, when ρ = 1 and τ = ∞, (8) behaves like the drift process of the diffusion identified in Proposition 1. That is, the process in (8) behaves as if the number of streaming flows is fixed. To loosely summarize Proposition 2, the number of file transfers in the system can be approximated as t κL κL n( ) + o( ). (9) N L (t) ≈ N L (0) + 1 − ρL τ L 1 − ρL Next we refine this by obtaining a diffusion approximation.

Proposition 3 (Diffusion Limits). Assume that (A1-A4) hold. Assume also that τ < ∞. √ ¯ L = κL ρLL , M ¯ L = κ , sL Consider (4) for each of the systems indexed by L with N κL , τ L = m η (1−ρ )η p ˆ L (·), Nˆ L (·)) as defined in (4). If (M ˆ L (0), Nˆ L (0)) ⇒ µC L τ L . Consider (M as in (A4) and sL n = (m(0), ˆ n ˆ (0)) , then ˆ L (·), N ˆ L (·)) ⇒ (m(·), (M ˆ n ˆ (·)), (10) where the diffusion limits (m(·), ˆ n ˆ (·)) satisfy Z t √ m(u)du ˆ + 2τ W1 (t) m(t) ˆ = m(0) ˆ − τη 0 Z t p [ηρm(s) ˆ − ηˆ n(s)] ds + 2ρ W2 (t) n ˆ (t) = n ˆ (0) +

(11)

0

for all t ≥ 0, where W1 (·) and W2 (·) are independent standard Brownian motions. The proof of Proposition 3 is postponed to Section 6. Some comments on Proposition 3 are in ˆ L (·), Nˆ L (·)) are centered at the equilibrium of the fluid model, properly order. First, note that (M scaled. Second, (11) is a so-called affine drift diffusion. A lot is known about these diffusions, and in Proposition 4 we characterize the distributions associated with this process. Also, as expected the m ˆ component behaves like an Orstein-Uhlenbeck process as is expected from a M/M/∞ queue. Third, the time scale separation between m and n is present in the diffusion limit as well. A Rt √ conjecture that suggests itself when τ is infinite is that n ˆ (t) = n ˆ (0) − η 0 [ˆ n(s)] ds + 2ρ W2 (t). That is, n ˆ behaves like a Orstein-Uhlenbeck process and the impact of the m ˆ component has simply averaged out. We leave this as a conjecture in this paper.

7

Proposition 4. If (m(0), ˆ n ˆ (0)) is a Gaussian random variable then (m(·), ˆ n ˆ (·)) is a Gaussian process that can be completely characterized given the mean and covariance of (m(0), ˆ n ˆ (0)). Furthermore, if (m(0), ˆ n ˆ (0)) has zero mean and covariance V , which is defined in (12) below, then (m(·), ˆ n ˆ (·)) is a stationary Gaussian process with the marginal distribution of (m(t), ˆ n ˆ (t)) being zero mean Gaussian with covariance V . Here V is given by # " 1 ρ V =

Proof

η

η(τ +1) ρ2 η(τ +1)

ρ η(τ +1)

+

ρ η

.

(12)

The proof follows Section 5.6 of [5]. We have an affine drift diffusion of the form dX(t) = AX(t)dt + σdW (t),

where A=



−τ η 0 ηρ −η



,σ =

 √

2τ 0

0 √ 2ρ



and W (t) =



W1 (t) W2 (t)



.

From Section 5.6 of [5], we know that if X(0) is Gaussian with mean λ(0) and covariance matrix V (0) then X(t) is a Gaussian process with mean λ(t) = Φ(t)λ(0) and covariance function # " Z V s

ρ(s, t) = Φ(s) V (0) +

t

Φ−1 (u)σσ T Φ−T (u)du ΦT (t),

0

˙ where Φ(t) satisfies Φ(t) = AΦ(t) with Φ(0) = I. The functions λ(·) and ρ(·, ·) completely characterize the Gaussian process. Furthermore, from Theorem 6.7 of [5], X(·) is a stationary, zero mean Gaussian process if λ(0) = 0, and V (0) = V where AV + V AT = −σσ T . Solving for V we obtain the result. Proposition 4 allows us to come up with the following stationary approximation that is valid for large L and t. ML ≈ NL ≈

κL √ L + κ X1 η

√ κL ρL κL + X2 , (1 − ρL )η (1 − ρL )

 X1 is a zero mean Gaussian random variable with covariance V given by (12). X2 Finally, we note that the asymptotic analysis carried out in this section applies, essentially without modification, to weighted allocation rules that share capacity between streaming flows and file transfer by weighting each streaming flow as if it were θ flows in the fair sharing rule, for a design parameter θ > 0. To be specific, under such a rule, the model of (2) is modified only as Cµn when n > 0. Such rules are known to have to good properties [3]. In this Q(m,n),(m,n−1) = θm+n ˆ L (·), N ˆ L (·)) in the same manner as described above. case, we obtain limits for (θ M where



(13)

8

5

Proof of Proposition 2

We break the proof into two parts. First we estimate the deviation of the nL from its fluid limit along “good paths.” Then we estimate the probability that these good paths occur. Introduce the notations ∆i (s) = Ξi (s) − s. Equations (3) can be rewritten as Rt N L (s) N L (t) = N L (0) + ν L t − µC L 0 M L (s)+N L (s) ds   R
L t N (s) ds +∆1 ν L t − ∆2 µC L 0 M L (s)+N L (s) Rt M L (s) L L L L = N (0) − µC (1 − ρ )t + µC 0 M L (s)+N ds   L (s) R
L (s) t N +∆1 ν L t − ∆2 µC L 0 M L (s)+N L (s) ds .

(14)

Recall from Proposition 2 the notation nL (t) =

1 L L N (τ t), sL n

mL (t) =

1 M L (t), κL

L L L L L L 2 where sL n = κ /(1 − ρ ), and τ = κ /[µC (1 − ρ ) ]. The initial conditions N L (0), M L (0) are such that, in probability, nL (0) → n(0) and M L (0)/κ → m(0), for some finite, strictly positive limits n(0). It follows directly from (14) that

Rt mL (τ L u) nL (t) = nL (0) − t + 0 (1−ρL )m L (τ L u)+nL (u) du     √ 2  L L 1−ρ κ 1 L ρ t + √ L √ L ∆1 1−ρL κ κ    √ 2 R   L (u) L L t n 1−ρ 1 κ − √ L √ L ∆2 0 (1−ρL )mL (τ L u)+nL (u) du . 1−ρL κ

κ

We introduce further notations:   √    L 2 L  1−ρ κ 1 L L  ǫ (t) = √ √ ∆1 ρ t   1−ρL κL κL     L  √ 2 R t nL (u) 1−ρ κL 1 √ √ − L ∆2  0 (1−ρL )mL (τ L u)+nL (u) du .  1−ρL κ κL    L n ˜ (t) = nL (t) − ǫL (t). The previous equation may thus be rewritten Z t mL (τ L s) n ˜ L (t) = n ˜ L (0) − t + ds. L L L L ˜ L (s) 0 (1 − ρ )m (τ s) + ǫ (s) + n

(15)

Now we turn our attention to identifying the “good paths” and the deviation of nL from n 1 δ′ ′ ′′ on these paths. Let β = 12 min(m(0), 1/η, 1), δ = βρ 4 , δ = 2 min(n(0), δ), δ = 2 and B = max(m(0), 1/η) be fixed positive constants.

9

Fix a T > 0. For a given L large enough such that ρL ≥ 2δ/β, we say a path is good if along the path we have the following for some constant ǫ 0, and all λ > 0, it holds that P(sup0≤t≤T |Ξ(t) − t| ≥ λT ) ≤ e−T h(λ) + e−T h(−λ) ,

(20)

h(λ) := (1 + λ) log(1 + λ) − λ

(21)

where is the Cram´er transform of a unit mean, centered Poisson random variable. In the above formula, it is understood that h(−λ) = +∞ if λ > 1. √ If we let λ depend on T , under the assumption that λ(T ) T → +∞ as T → ∞, we have that lim P(sup0≤t≤T |Ξ(t) − t| ≥ λ(T )T ) = 0.

T →∞

Proof: These results and their proofs are standard (see e.g. [10]). The first part is established by bounding the left-hand side by the sum of the two terms obtained upon replacing |Ξ(t) − t| in its expression by Ξ(t) − t and t − Ξ(t) respectively, and then bounding each of this two terms by using Doob’s inequality, applied to the two non-negative submartingales M± (t) := exp(±θ(Ξ(t) − t)), and finally optimising over the positive parameter θ. The second statement is a straightforward consequence of the first. √ Let T > 0 be fixed. Recall that under the assumptions (A1-A4), the term κL /(1 − ρL ) goes to +∞ as L → ∞. It follows readily from Lemma 1 that P( sup |ǫL (t)| > ǫ) → 0 as L → ∞, 0≤t≤T

11

(22)

for every constant ǫ > 0. By assumption, we also have P(|nL (0) − n(0)| > ǫ) → 0 as L → ∞. To obtain the probability estimates for the mL components, we divide the arguments into two cases. First we consider the case where τ < ∞. In this case, the first part of the proposition, combined with the fact that the limiting process m(·) is Lipschitz-continuous yields P( sup |mL (τ L t) − m(τ t)| > ǫ) → 0 as L → ∞,

(23)

0≤t≤T

RT for every constant ǫ > 0. Since 2β ≤ m(t) ≤ B for all t ∈ [0, T ], we have P( 0 1mL (τ L t) RT ǫ) → 0 and P( 0 |mL (τ L t) − m(τ t)| dt > ǫ) → 0 as L → ∞. This along with (22,23) implies from the previous arguments leading upto (19) that P( sup |nL (t) − n(t)| > K ′ ǫ) → 0 as L → ∞,

(24)

0≤t≤T

for every constant ǫ > 0, proving the proposition. Now we turn our attention to the case when τ = ∞. The random variable M L (t) is distributed as the sum of two independent random variables, one binomial with parameters (M L (0), e−ηt ) and one Poisson random variable with parameter (κ/η)(1 − e−ηt ). Thus, its expectation and variance read κ κ E(M L (t)) = M L (0)e−ηt + (1 − e−ηt ), var(M L (t)) = M L (0)e−ηt (1 − e−ηt ) + (1 − e−ηt ). (25) η η Note also that the Chernoff bounds obtained from such a random variable will be more stringent than those for a corresponding Poisson random variable with the same mean. We thus obtain R τ LT RT P(M L (t) ≤ κL /(2η))dt E 0 1mL (τ L u)≤β du = τ1L h0 i R τ LT ≤ τ1L η −1 + 1/η P(M L (t) ≤ κL /(2η))dt

which converges to zero because for t ≥ η −1 , EM L (t) ≥ (κL /η)(1 − 1/e) > κL /(2η) and we see that the integrand above is exponentially small in κL . Now consider o p R T 1 L L R τ L T n 1 1 1 1 L E 0 κL M (τ u) − η du ≤ τ L 0 η − κL E(M (t)) + κ1 var(M L (t)) dt o p p R τ L T n −ηt M L (0) 1 1 1 L L √ M (0)/κ , 1/η dt e − η + L max ≤ τL 0 κL p κ  p 1 M L (0) 1 T ≤ τ L ηκL − η + √ L max M L (0)/κ, 1/η , κ

where we have used the expressions (25). This last term converges to zero because both τ L and κL RT RT tend to infinity as L → ∞. Therefore we have P( 0 1mL (τ L t) ǫ) → 0 and P( 0 |mL (τ L t) − m(τ t)| dt > ǫ) → 0 as L → ∞ completing the proof as before.

6

Proof of Proposition 3

ˆ R2 ) denote the space of all continuous functions f : IR2 → IR that have finite limits at Let C(I ˆ R2 ) that have continuous first and second infinity and let Cˆ 2 (IR2 ) denote the space of all f ∈ C(I 12

partial derivatives. The generator A of the process in (11) is given by Af (m, n) = −ηmτ

∂f (m, n) ∂ 2 f (m, n) ∂f (m, n) ∂ 2 f (m, n) +τ + [ηρm − ηn] + ρ . ∂m ∂m2 ∂n ∂n2

(26)

ˆ R2 )} is single Theorem 8.1.6 of [4] ensures that the closure of {(f, Af ) : f ∈ Cˆ 2 (IR2 ), Af ∈ C(I 2 ˆ R ). valued and generates a Feller continuous semigroup on C(I L The generator of the L-th system A is defined in the original chain by, suppressing the superscript L on the rates, AL f (m, n) = κ [f (m + 1, n) − f (m, n)] + mη [f (m − 1, n) − f (m, n)] Cµn1n>0 [f (m, n − 1) − f (m, n)] . +ν [f (m, n + 1) − f (m, n)] + m+n ¯

¯

−1 ˆ L (·), N ˆ L (·) is a Markov chain on (m, ,···} The centered and scaled process (M ˆ n ˆ ) where m ˆ ∈ {− sMm , − Msm ¯

¯

, · · · }, where we have suppressed the superscript L on the quantities for simand n ˆ ∈ {− sNn , − Ns−1 n

plicity of notation. Note that sm , sn , ¯ −1 ¯ ,···} {− sMm , − Msm

¯ ¯ {− sNn , − Ns−1 , · · · }. n

¯ M sm ,

¯ N sn

→ ∞ as L → ∞ by (A1-A4). Let EL denote

× The generator of centered and scaled chain is given by     1 L ˆ ,n ˆ − f (m, ˆ n ˆ) A f (m, ˆ n ˆ ) = κτ f m ˆ + sm     1 ¯ + sm m)ητ +(M ˆ f m ˆ − ,n ˆ − f (m, ˆ n ˆ) sm     1 +ντ f m, ˆ n ˆ+ − f (m, ˆ n ˆ) sn   ¯ + sn n ¯ /sn }   1 Cµτ · (N ˆ ) 1{ˆ n > −N − f (m, ˆ n ˆ) . f m, ˆ n ˆ− + ¯ + sm m ¯ + sn n sn M ˆ +N ˆ

(27)

The domain of the generator is B(EL ) which is the set of all bounded functions on EL . If f ∈ C 2 (IR2 ) then by Taylor’s formula, we have     1 ˆ n ˆ) 1 ∂f (m, ˆ n ˆ) 1 1 ∂ 2 f (m, 1 ,n ˆ = f (m, ˆ n ˆ) ± · + · + o f m ˆ ± sm ∂m ˆ sm 2 ∂ m ˆ2 s2m s2m
¯ whenever m ˆ > − sMm , and as usual the notation o s12 denotes a function such that s2 o s → ∞. Similarly,     1 ˆ n ˆ) 1 1 ∂ 2 f (m, ∂f (m, ˆ n ˆ) 1 1 · + · 2 +o 2 = f (m, ˆ n ˆ) ± f m, ˆ n ˆ± 2 sn ∂n ˆ sn 2 ∂n ˆ sn sn ¯

whenever n ˆ > − Ns−1 . n 13

1 s2




→ 0 as

Using this we can rewrite (27) as   ¯ + sm m)η)τ ∂f (m, ˆ n ˆ ) (κ − (M ˆ L ˆ A f (m, ˆ n ˆ) = ∂m ˆ sm  2   ¯ 1 ∂ f (m, ˆ n ˆ) (κ + (M + sm m)η)τ ˆ 2 2 + O(sm )o(sm ) + s2m 2 ∂m ˆ2   ¯ + sm m) ¯ + sn n ∂f (m, ˆ n ˆ ) [ν(M ˆ + (ν − Cµ)(N ˆ )]τ + ¯ + sm m ¯ + sn n ∂n ˆ (M ˆ +N ˆ )sn  2   ¯ + sm m) ¯ + sn n 1 ∂ f (m, [ν(M ˆ + (ν + Cµ)(N ˆ )]τ ˆ n ˆ) 2 2 + O(sn )o(sn ) , (28) + ¯ + sm m ¯ + sn n 2 ∂n ˆ2 (M ˆ +N ˆ )s2n ¯

¯

ˆ > − Ns−1 and as usual O(s) denotes a function such that 1s O(s) when f ∈ Cˆ 2 (IR2 ) and m ˆ > − sMm , n n is bounded for all s. ¯ ,N ¯ and τ in (28), for f ∈ Cˆ 2 (IR2 ), we obtain by algebraic Substituting the values of sm , sn , M manipulation that ∂f (m, ˆ n ˆ) AˆL f (m, ˆ n ˆ ) = −ητ m ˆ ∂m ˆ     ˆ n ˆ) mτ ˆ 1 1 ∂ 2 f (m, √ 2τ + + O(κ)o + 2 ∂m ˆ2 κ κ " # ρm ˆ −n ˆ ∂f (m, ˆ n ˆ) + 1−ρ 1 ∂n ˆ √ m ˆ + √nˆ η + k k " ρ     m ˆ n ˆ # 1 ∂ 2 f (m, ˆ n ˆ ) 2 η + (1 − ρ)ρ √k + (1 + ρ) √k (1 − ρ)2 κ + o +O . 1 ˆ m ˆ √ √n 2 ∂n ˆ2 (1 − ρ)2 κ η + (1 − ρ) k + k ˆ R2 ) As L → ∞, we have, for all f ∈ Cˆ 2 (IR2 ) with Af ∈ C(I ˆL sup A f (m, n) − Af (m, n) → 0, {(m,n)∈IR2 }

where Af is given by (26). The convergence pointwise is immediate from (A1-A4). The convergence ˆ R2 ). Therein the sup norm follows from the fact that we only need to consider functions Af ∈ C(I ˆ L (0), Nˆ L (0)) fore, using Theorems 4.2.11 and 2.6.1 of [4] and assuming that the initial conditions (M converge to some random variables (m(0), n(0)) in distribution completes the proof.

References [1] E. G. Coffman, Puhalskii, A. A. and Reiman, M. I., “Polling Systems in Heavy Traffic: A Bessel Process Limit,” Math. of OR, 23(2), pp.257-304, 1998. [2] C. Courcoubetis, A. Dimakis and M.I. Reiman. “Providing Bandwidth Guarantees over a Best-effort Network: Call-Admission and Pricing. IEEE INFOCOM, 2001. 14

[3] S. Deb, Ganesh, A. and Key, P., “Resource Allocation between Persistent and Transient Flows,” INFOCOM, 2004. [4] S. N. Ethier and Kurtz, T. G., Markov Processes: Characterization and Convergence, Wiley, 1986. [5] I. Karatzas and Shreve, S. E., Brownian Motion and Stochastic Calculus, Second Edition, Springer, 1991. [6] P. Key, L. Massouli´e, A. Bain and F. Kelly, “Fair Internet traffic integration: network flow models and analysis,” Annals of Telecommunication 59 (11-12), pp.1338-1352, 2004. [7] T. Kurtz. Approximation of Population Processes, vol. 36, CBMS-NSF Regional Conference Series in Applied Mathematics, 1981. [8] A. Mandelbaum, Massey, W. A. and Reiman, M. I., “Strong Approximations for Markovian Service Networks,” Queueing Systems, 30, pp.149-201, 1998. [9] L. Massouli´e and J. Roberts, Bandwidth sharing and admission control for elastic traffic, Telecommunication Systems, 15, p.185-210, 2000. [10] G. Shorak and J. Wellner, Empirical processes with applications to statistics, Wiley, New York, 1986. [11] K. Yamada, ‘Diffusion Approximations for Storage Processes with General Release Rules,” Math. of OR, 9(3), pp.459-470, 1984.

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