An Upper Bound on Overflow Probability in ... - UNC Chapel Hill

An Upper Bound on Overflow Probability in Transient Source Systems Thomas F. Reid∗

Vidyadhar G. Kulkarni†

September 6, 2005

Abstract In this paper we consider a single buffer fluid model where sources arrive according to a Poisson process, and stay in the system for independent and identically distributed (iid) exponential amounts of time and then depart. While they are in the system, they generate fluid traffic that is modulated by iid finite state continuous time Markov chains (CTMC). The fluid in the buffer is drained at a constant rate. We study stability of the buffer content process, and then derive an exponential upper bound for the tail of the buffer content process in steady state. The results have applications in design and operation of ATM switches: we use them to develop call admission policies, and also for capacity design. Numerical results are presented to demonstrate the methodology.

∗ †

Air Force Institute of Technology University of North Carolina at Chapel Hill

1

1

Introduction

The growing interest in the use of asynchronous transfer mode (ATM) for broadband integrated services digital networks (B-ISDN) has fueled a large body of research in recent years. The basic problem is to increase efficiency and resource utilization in such networks while guaranteeing a given quality of service (QoS). Frequently, the QoS is given as a maximum acceptable probability of packet loss (or cell loss probability). The vast majority of research in this area, including the well-known effective bandwidth (EBW) results, assumes that traffic sources, once admitted, never leave the system. That is, they assume permanent sources. Beginning with simple on-off sources in [AMS82], the effective bandwidth methodology has been expanded to include multiple-state sources [EM93], non-homogeneous sources [KGC], general ergodic sources [KWC93], and sources controlled by Markov regenerative processes [Kul96]. De Veciana et al. [dVKW95] provides an excellent overview of the current literature in EBW-based admission control, while Kesidis et al. [KWC93] provide a unified framework for effective bandwidth using the large deviations approach for a broad class of environment processes. In this paper, we explicitly model a transient source system: i.e., a situation where sources arrive according to a random process, stay in the system for a random amount of time, and then depart. While they are in the system they generate traffic according a rate that is modulated by a random environment process. The traffic generated from the sources in the system is fed into a single buffer, which removes the traffic at a fixed rate. We seek an upper bound on the probability of overflow from this buffer. Duffield and Daley [DD] propose such a bound for simple constant-bit-rate sources, but their analysis is incomplete. We derive a bound for the more general case where the source output rate is governed by a continuous time Markov chain (CTMC) environment process. The paper is organized as follows. Section 2 presents relevant existing results about 2

single buffer stochastic fluid flow systems. In particular, we restate the Palmowski-Rolski (PR) bounds on overflow probabilities from Palmowski and Rolski [PR94] and Gautam and Kulkarni [GK97]. We also extend them to the case of multiplexed sources. We describe the model of the transient source system in Section 3, and derive the stability condition for it. In section 4 we introduce a system with a fixed number of sources that will be useful in the analysis of the transient source system. We show that this permanent source system converges to (in a sense made precise in Section 4) the transient source system as the number of sources becomes large. In Section 5 we derive the PR bounds for the permanent source system and study their limit as the number of sources becomes large. This limit produces the TS (Transient Source) bounds for the transient source system. Using the TS bounds we propose a congestion control scheme in Section 6. This scheme is then compared with the standard effective bandwidth scheme. In Section 7 we consider the design problem: find the minimum output rate from the buffer to guarantee the QoS. Numerical and graphical comparison is reported in Section 8. It shows that, as expected, the effective bandwidth scheme is reasonable only when the sources stay in the system for sufficiently long periods of time.

2

Preliminary Results

Consider a single-buffer fluid flow model with infinite buffer capacity, where the input traffic is modulated by a random environment process {Z(t), t ≥ 0} with state space S; i.e., when the environment state is k ∈ S, the input traffic arrives at rate r(k). Let R = [R(i, j)]i,j∈S be a diagonal matrix with R(k, k) = r(k), k ∈ S. We assume that {Z(t), t ≥ 0} is a finitestate irreducible CTMC with infinitesimal generator matrix Q with stationary distribution

3

π = [π(i)]i∈S , satisfying X

πQ = 0 and

π(i) = 1.

i∈S

The input traffic accumulates in a buffer from which it is removed at a constant rate c. Let X(t) be the buffer content at time t. The dynamics of the buffer content process are given by   

r(Z(t)) dX(t) =  dt  (r(Z(t)))+

if X(t) > 0 if X(t) = 0

where (x)+ = max(x, 0). Define the steady state drift d to be d=

X

π(i)r(i) − c.

i∈S

It is known that {X(t), t ≥ 0} is stable, i.e., X(t) has a limiting distribution as t → ∞, if d < 0 [PR94, GK97]. We shall assume that the above stability condition is satisfied. Let η ≥ 0 be such that the largest positive eigenvalue of R + η1 Q is c, and let h be the corresponding eigenvector, i.e., h satisfies 1 h(R + Q) = ch . η (That such an η exists is known, see Elwalid and Mitra [EM93].) The following theorem is from Palmowski and Rolski [PR94], as modified by Gautam and Kulkarni [GK97]. Theorem 1 (PR bound) P(X > x) ≡ lim P(X(t) > x) ≤ ge−ηx t→∞

where g≡

hh, 1i . min h(k)/π(k)

k∈S:r(k)>c

4

Here, h·, ·i denotes the vector inner product, and 1 is a vector of ones. The PR bound theorem can be generalized to handle multiple input sources as follows. Suppose there are N independent input sources, the ith source being modulated by an irreducible CTMC {Zi (t), t ≥ 0} with a finite state space S ( i) and infinitesimal generator Q( i). When Zi (t) = k, the ith source generates traffic at rate r( i)(k). As before, let R( i) = [R( i)(j, k)]j,k∈S be a diagonal matrix with R( i)(k, k) = r( i)(k). Let π ( i) be the stationary distribution of {Zi (t), t ≥ 0}. The above system of N sources can be reduced to a system with a single aggregate source by using the notation and methods of Elwalid and Mitra [EM93]. We can think of Z [a] (t) = (Z1 (t), Z2 (t), . . . , ZN (t)) as the state of a single aggregate environment process at time t. (We use the superscript [a] to denote quantities associated with the aggregate n

o

system.) Z [a] (t), t ≥ 0 is an irreducible CTMC with state space S [a] = S (1) × S (2) × · · · × S (N ) . Now let A ⊗ B denote the Kronecker product of A and B. It is obtained by replacing each element a(i, j) of A by a(i, j)B. Let ⊕ denote the Kronecker sum: A⊕B = A⊗I n +I m ⊗B, where A is (m×m), B is (n×n) and I k is an identity matrix of order k. Using this notation, n

o

the infinitesimal generator of Z [a] (t), t ≥ 0 can be written as Q[a] = Q(1) ⊕ Q(2) ⊕ · · · ⊕ Q(N ) . n

(1)

o

The stationary distribution Z [a] (t), t ≥ 0 is given by π [a] = π (1) ⊗ π (2) ⊗ · · · ⊗ π (N ) .

(2)

R[a] = R(1) ⊕ R(2) ⊕ · · · ⊕ R(N )

(3)

Furthermore,

5

is the diagonal input-rate matrix of the aggregate system. Note that, for k = (k(1), k(2), ..., k(N )) ∈ S [a] , we have R[a] (k, k) = r(k) =

N X

r(i) (k(i)).

i=1

The following result, due to Elwalid and Mitra [EM93], gives η and the associated eigenvector h for the aggregate-source system via the solution of N smaller, coupled eigenvalue problems, which are easier to deal with computationally. Theorem 2 (EM93): Let ebi (η) be the largest (most positive) eigenvalue of (R(i) + η1 Q(i) ), and let η and h(i) (η) (i = 1, . . . , N ) satisfy (i)



(i)

ebi (η)h (η) = h (η) R

(i)

+

1 (i) Q η

PN

i=1 ebi (η) = c.

and

   

(4)

 

Then η is such that the largest eigenvalue of (R[a] + η1 Q[a] ) is c and h[a] ≡ h(1) ⊗ h(2) ⊗ · · · ⊗ h(N )

(5)

satisfies 1 h[a] (R[a] + Q[a] ) = ch[a] . η Using Theorem 2, the PR bound in Theorem 1 can be generalized to systems with N independent sources as follows. Theorem 3 Let η and h(i) be as in Theorem 2. Then N Y

P (X > B) ≤

hh[a] , 1i

i=1 N Y h(i) (k(i))

min

k∈S [a] :r(k)>c i=1

6

π (i) (k(i))

e−ηB .

(6)

Proof.

Let k = [k(1), k(2), . . . , k(N )] ∈ S [a] . Equation (5) implies that N Y

h[a] (k) =

h(i) (k(i))

(7)

i=1

so N Y

hh[a] , 1i =

hh(i) , 1i .

i=1

Similarly, Equations (2) and (5) imply N h(i) (k(i)) h[a] (k) Y = . π [a] (k) i=1 π (i) (k(i))

(8)

Now, substituting in Theorem 1, we get P (X > B) ≤

hh[a] , 1i h[a] (k) min P [a] k∈S [a] : r (i) (k(i))>c π (k)

e−ηB .

Substituting the results of Equations (7) and (8) gives Equation (6). When sources are independent and stochastically identical (iid), Equation (6) simplifies considerably. A different representation of the state space S [a] will simplify the discussion and analysis. Let all sources be independent and identical with common state space S (k) = S, common generator matrix Q(k) = Q (hence π (k) = π) and common rate matrix R(k) = R for k = 1, . . . , N . Let S =| S |. Now let σ(j) denote the number of sources in state j, and define Σ = {σ ≡ (σ(1), σ(2), . . . , σ(S)) ∈ Z S : hσ, 1i = N }. We extend (or, perhaps, abuse) the notation and denote the total output rate when in state σ by r(σ) ≡

PS

j=1

r(j)σ(j).

Theorem 4 Let η and h be as defined in Theorem 2. If Q is irreducible, and hπR, 1i − c/N < 0

(9)

P(X > x) ≤ ge−ηx

(10)

then

7

where (hh, 1i)N

g≡



QS

min

i=1

σ∈Σ:r(σ)>c

 h(i) σ(i) π(i)

n

.

(11)

o

Here, S =| S | and Σ = σ ∈ Z S : hσ, 1i = N, σ ≥ 0 . Proof.

Since all of the sources are identically distributed, for any given value of η all of

the eigenvalue problems of Theorem 2 share the same solution. That is, eb(i) (η) = eb(η) and h(i) = h for i = 1, . . . , N . Note that this η and h satisfy Equation (4), so that eb(η) = c/N . From Theorem 3 we get N Y

hh(i) , 1i =

i=1

N Y

hh, 1i

i=1

= (hh, 1i)N ,

(12)

and N Y h(i) (k(i)) i=1

π (i) (k(i))

N Y h(k(i))

=

i=1 S Y

=

j=1

π(k(i)) h(j) π(j)

!σ(j)

.

(13)

Substituting Equations (12) and (13) into Equation (6) yields the desired result.

In the next section, we introduce the general transient source (TS) system and then show how we can use the PR bound of Theorem 4 to obtain a bound for the TS system.

8

3

Transient Source System (TS): Stability Condition

Consider a fluid flow model of a single buffer where traffic sources arrive according to a P P (λ). They stay for an Exp(µ) random time and then depart. During their sojourn in the system they behave like independent and stochastically identical sources modulated by iid CTMCs each with finite state-space S, generator matrix Q and traffic rate matrix R. At the time of arrival, the source (i.e., its modulating environment) is in state k ∈ S with probability a(k). Let a ≡ [a(k)]k∈S . The generated traffic is input into an infinite buffer and is removed from that buffer at constant rate c. Let L(t) be the number of sources in the system at time t. Clearly, {L(t), t ≥ 0} is the queue length process of an M/M/∞ system. Let Li (t) be the number of sources in state i at time t. Then we can represent the state of all sources in the system at time t by (L1 (t), L2 (t), . . . , LS (t)). Now let X(t) be the buffer content at time t. The dynamics of the buffer content process is described by   

S if X(t) > 0 dX(t) i=1 Li (t)r(i) − c = hP i+  S dt  if X(t) = 0. i=1 Li (t)r(i) − c

P

Thus, the state of the system at time t is described by {(X(t), L1 (t), . . . , LS (t)) , t ≥ 0}. We call such a system a TS system. It is completely characterized by {a, Q, R, λ, µ, c}. The next theorem states the stability condition for the TS system. Theorem 5 The transient source system is stable if λha (µI − Q)−1 R, 1i < c . Proof.

Let τ be the expected total traffic generated by a typical source during its sojourn

in the system. Since the sources arrive at rate λ, the long run rate at which traffic enters the buffer is given by λτ . The buffer content can be emptied at a maximum rate of c. Hence, 9

the TS system will be stable if λτ < c.

(14)

Next we compute τ . Suppose a source enters the system at time 0 and leaves at time T . It is modulated by the CTMC {Z(t), t ≥ 0}. Let Pi,j (t) = P (Z(t) = j|Z(0) = i),

i, j ∈ S.

The expected total traffic generated by the source is given by τ = E

Z

!

T

r[Z(t)]dt

=

0

=

Z

∞

µe

t=0 Z ∞ u=0

=

Z

∞

Z

∞

E(r[Z(u)])dudt

µe−µt dtE(r[Z(u)])du

t=u

e−µu

S S X X

r(j)P (Z(u) = j|Z(0) = i)P (Z(0) = i)du

j=1 i=1

S Z S X X

∞

j=1 i=1 u=0

=

t

u=0

u=0

=

Z

−µt

S X S X

e−µu r(j)Pij (u)a(i)du

a(i)r(j)

Z 0

j=1 i=1

∞

e−µu Pij (u)du

The integral in the last equation is the Laplace transform (evaluated at µ) of the transition probability, and is given by (µI − Q)−1 ij (see [Kul95]). Thus, τ =

S X S X

a(i)r(j)(µI − Q)−1 ij

j=1 i=1

= ha (µI − Q)−1 R, 1i.

(15)

This yields the Theorem. We are interested in finding lim P (X(t) > x),

t→∞

assuming that the system is stable. Unfortunately, the complexity of the TS system makes direct analysis of the this overflow probability intractable. Hence, we consider a special 10

sequence of permanent source systems, P S(N ), N = 1, , 3, ..., in the next section. This sequence is designed so that a P S(N ) “converges” to TS as N tends to infinity, in a manner made precise in the next section.

4

A Permanent Source System P S(N )

Consider a system, denoted by P S(N ), with N permanent sources, each of which alternates between the set of active states S = {1, 2, . . . , S} and an additional dormant state, s˜. When active, a source behaves just as a source in the TS system except that, instead of leaving the system (with rate µ), it moves to the dormant state and produces no traffic. After spending an Exp(θN ) amount of time in the dormant state, the source becomes active again in state i with probability a(i), i = 1, 2, 3, .., S. Thus, these modified permanent Markovian sources ¯ ≡ S S{˜ are independent and identically distributed, with state space S s} and traffic rate matrix S 

¯ ≡ R

s˜ 

S R 0  . s˜ 0 0

(16)

The infinitesimal generator for each source depends on N , and is given by S 

¯N ≡ Q

S  Q − µI s˜



θN a

s˜ 

µ1  −θN

.

(17)

We shall denote this permanent source system as PS(N ); the quantities {a, Q, R, µ, c, N, θN } characterize this system. For a PS(N ) system, let L(N ) (t) be the number of sources in the (N )

active states, Li (t) be the number of sources in state i, and XN (t) be the buffer content, at time t. Our first result is given by 11

Theorem 6 Suppose lim N θN = λ. Then, as N → ∞, N →∞

D

{XN (t), t ≥ 0} → {X(t), t ≥ 0} , D

where {X(t), t ≥ 0} is the buffer content process in the TS system, and “→” denotes weak convergence (i.e., convergence of all the finite-dimensional distributions),. The proof proceeds by several lemmas, the first from [Sti76]. It is restated below for completeness. Consider a machine shop with N machines and N servers. The up times of the machines are iid with common distribution G(·) and down times are iid with common distribution FN (·). Let D(N ) (t) be the number of down machines at time t. {D(N ) (t), t ≥ 0} is the queue length process in a GI/GI/N/N/N system. Let D(t) be the number of customers in an M/G/∞ queue with arrival rate λ and service time distribution G(·). Then we have Lemma 1 (Stidham) Suppose that, for each t ≥ 0, FN (t) → 0 and N FN (t) → λt n

o

(18) (19)

D

as N → ∞. Then D(N ) (t), t ≥ 0 → {D(t), t ≥ 0}. Using this result, we can show Corollary 1 Let L(N ) (t) be the number of sources in the active states at time t in the PS(N ) system with parameters {a, Q, R, µ, c, N, θN }. Let L(t) be the queue length process in a TS system with parameters {a, Q, R, µ, λ, c}. If N θN → λ as N → ∞ then n

o

D

L(N ) (t), t ≥ 0 → {L(t), t ≥ 0} . 12

(20)

Proof.

To apply Stidham’s result, we interpret “down” as the set of active states, and

“up” as the dormant state, s˜. Then FN (t) = 1 − e−θN t , so that, with N θN → λ, we get lim FN (t) = lim 1 − e−λt/N = 0

N →∞

N →∞

which satisfies Equation (18). Also, 1 − e−λt/N lim N FN (t) = lim N →∞ N →∞ 1/N λtN −2 e−λt/N = lim (by L’Hospital’s rule) N →∞ N −2 = λt satisfies Equation (19). L(N ) (t) is the number of customers in an M/M/N/N/N system, corresponding to D(N ) (t) in Lemma 1. L(t) is the number of customers in an M/M/∞ system and corresponds to the queue length D(t) in the M/G/∞ queue of that lemma. Thus, the number of active sources in a PS(N ) system weakly converges to the number of sources in a TS system. The next corollary states (the proof is omitted) the weak convergence (N )

(N )

(N )

result for the vector process {(L1 (t), L2 (t), . . . , LS (t)), t ≥ 0} . Corollary 2 Suppose Equation (20) holds. Then (N )

(N )

D

(N )

{(L1 (t), L2 (t), . . . , LS (t)), t ≥ 0} → {(L1 (t), L2 (t), . . . , LS (t)), t ≥ 0}.

(21)

The proof of our main result follows the approach used by Kulkarni and Rolski in [KR94, Theorem 2.2], which relies on a theorem by Borovkov [Bor84, Theorem 4.1.3]. This idea is alluded to in [AMS82], who attribute the model to Kosten [Kos74]. Proof of Thm 6.

Define ζN (t) ≡

S X

(N )

r(i)Li (t) − c,

i=1

13

and ζ(t) ≡

S X

r(i)Li (t) − c.

i=1

Then the buffer content process in the P S(N ) system satisfies    ζN (t)

if XN (t) > 0 d XN (t) =  dt  (ζ (t))+ if X (t) = 0, N N while the buffer content process in the TS system satisfies    ζ(t)

if X(t) > 0 d X(t) =  dt  (ζ(t))+ if X(t) = 0 Let the initial condition of the systems be given by XN (0) = X(0) = x0 . Let Y (t) = x0 +

Rt 0

ζ(s)ds and YN (t) = x0 +

Rt 0

ζN (s)ds. By standard arguments (see, for example,

Borovkov [Bor76, p. 24]), we get X(t) = Y (t) − inf (0, Y (u)) 0≤u≤t



=

sup

Y (t),

Z

t



ζ(s)ds .

u

0≤u≤t

A similar equation holds for XN (t). From Corollary 2 we get D

ζN (t) → ζ(t). Since convergence in distribution is necessarily uniform on any closed interval of continuity [Doo90, p. 9], then, for each u, D

sup (ζN (t) − ζN (v)) → sup (ζ(t) − ζ(v))

−u≤v≤0

(22)

−u≤v≤0

which is condition (IV) of [Bor84, Theorem 4.1.3]. As in [KR94], condition (III) is satisfied by noting that E(

S X

(N )

r(i)Li (t))+ → E(

i=1

S X i=1

14

r(i)Li (t))+ .

Application of Borovkov’s Theorem 4.1.3 now gives the desired result. The next section discusses how we can use these results to provide a bound for the probability of overflow in a TS system.

5

Overflow Probability Bound: From P S(N ) to TS.

Consider a P S(N ) system with parameters {a, Q, R, µ, c, N, θN } with θN = λ/N . From the results in the previous section, we know that this P S(N ) system converges to a TS system with parameters {a, Q, R, µ, c, λ} as {N → ∞}. In this section we shall first derive the PR bounds for the P S(N ) system, which, in the limit as {N → ∞} will provide the corresponding bounds for the TS system. ¯ N be an (S + 1) by (S + 1) matrix as defined in Equation (17). Let w ¯ N satisfy Let Q ¯ N = 0. That is, w ¯ N . Furthermore, ¯ NQ ¯ N is the non-normalized stationary distribution of Q w 

¯ + let eb(ηN ) be the largest positive eigenvalue of R

1 ¯ QN ηN



, ηN be such that eb(ηN ) = c/N ,

¯ N be the associated eigenvector. From Theorem 4 we get the following PR bound for and h the steady-state buffer content process of the P S(N ) system: ¯ N be as defined above. Then ¯ N and h Theorem 7 Let ηN , w P (XN > B) ≤ gN e−ηN B , where 

gN ≡ min

¯ N ,1i N hh ¯ N ,1i hw

QS+1  h¯ N (i) σ(i)

σ∈ΣN :r(σ)>c

i=1

(23)

,

(24)

w ¯N (i)

and ΣN = {σ ∈ RS+1 : σ ≥ 0, integer, hσ, ri > c, hσ, 1i = N }. 15

(25)

Proof.

The theorem follows from Theorem 4 by using the normalized probability distri-

¯ N /hw ¯ N , 1i. bution w Theorems 7 and 6 provide a potential method for bounding the probability of overflow in a TS system as shown in the following theorem: Theorem 8 Let X be the steady state buffer content in the TS system. Then P (X > B) ≤ ge−ηN B ,

(26)

where g = lim gN , N →∞

and η = lim ηN . N →∞

Proof.

Theorem 6 gives us that P (XN > B) → P (X > B) as N → ∞. The theorem

follows from taking limits on both sides of the Equation (23). The limits g and η exist since the eigenvalues and eigenvectors of a matrix are continuous functions of its elements. Next we discuss how to compute the limits of gN and ηN . Since any non-zero scalar ¯ N = [hN , N ]. The next multiple of an eigenvector is also an eigenvector, we choose to use h theorem discusses the limits of ηN and the corresponding eigenvector hN . Theorem 9 Let η be the smallest positive solution to 

 µI − ηR − Q

det 

−λa



−µ1  λ + ηc

=0

(27)

and h be such that #

"

1 −λ a. h R + (Q − µI) = η η

16

(28)

Then lim ηN = η

N →∞

and lim hN = h.

N →∞

¯ N is an eigenvector, we can arbitrarily set h ¯ N = [hN , N ]. Then we Since h

Proof. 

¯N R ¯ + have h

1 ¯ QN ηN



=

c ¯ h , N N

Thus (ηN , (hN , 1)) is a solution to the generalized eigenvalue

problem 

 Q − µI

(hN , 1) 

µ1 

  = ηN (hN , 1) 

−λ

λa





c I N



−R 0  0

c

 .

(29)

Taking the limit as N → ∞, we see that (η, h) solves 

 Q − µI

(h, 1) 

λa

−λ







 −R 0 

µ1 

 = η (h, 1) 

0

c

 .

(30)

To calculate η, note that Equation (29) has a nontrivial solution if and only if 

 µI − Q + ηN

det 



c I N

−R

−λa



−µ1 λ + ηN c

  =0.

As stated in [EM93], ηN is the smallest positive solution. Taking the limit gives that η is the smallest positive solution to Equation (27). Equation (28) follows directly from Equation (30). Next we consider the limit of gN = AN /DN , where AN and DN are the numerator and the denominator of gN as in Equation (24). As stated in the proof of Theorem 9, we use ¯ N = [hN , N ]. Similarly, we use w ¯ N = [wN , N ] as the non-normalized the eigenvector h ¯ N . Thus stationary distribution of Q 

 Q − µI

0 = [wN , N ] 

θN a 17



µ1  −θN





=



w(Q − µI) + λa µhwN , 1i − λ

,

so wN solves wN (Q − µI) = −λa

(31)

and does not depend on N ! Hence we shall write wN = w. We shall first consider the limit of the denominator DN , which is given by DN = min σ ∈ΣN

S+1 Y

hN (i) wN (i)

i=1

!σ(i)

(32)

where ΣN is as defined in Equation (25). The following lemma gives a lower bound on DN as N → ∞. Lemma 2 Let h be as in Theorem 9 and w = wN be as given in Equation (31). Then  

h(i) lim DN ≥ min  N →∞ w(i) Proof.

 

!c/r(i)

: 1 ≤ i ≤ S, r(i) > 0 .

Since hN (S + 1) = wN (S + 1) = N , we have S Y

hN (i) wN (i)

DN = min σ ∈Σ i=1

!σ(i)

(33)

Since limN →∞ hN = h and wN = w, we have S Y

lim DN = min σ ∈Σ i=1

N →∞

h(i) w(i)

!σ(i)

.

Now let Σ∗ be as defined below: Σ∗ = {σ ∈ RS : σ ≥ 0, hσ, ri ≥ c.}. It is clear that (σ1 , σ2 , ..., σS+1 ) ∈ ΣN ⇒ (σ1 , σ2 , ..., σS ) ∈ Σ∗ . Hence, we get S Y

lim DN ≥ min∗ σ ∈Σ i=1

N →∞

18

h(i) w(i)

!σ(i)

.

(34)

Now, from Equation (30) we get h [ηR + (Q − µI)] = −λa. Subtracting from Equation (31) and rearranging, we get hηR = (h − w)(µI − Q), and hence h − w = hηR (µ(I) − Q)−1 . We note that the right hand side of this equation is positive, since the expected total traffic generated by a source, τ , is positive whenever R(i, i) > 0 for some i and Q is irreducible. Thus, Equation (15), with a = ηhR and letting R of Equation (15) be a matrix of zeroes except for R(i, i) = 1 for some given i, implies that each element of h − w is positive. Hence h > w, and each term of the above product is greater than one. Hence, we must choose the smallest possible values of σ(i), 1 ≤ i ≤ S without violating the constraint hσ, ri ≥ c. It is clear that (one can use the theory of linear programming to see this) the minimizing σ is of the following form for some 1 ≤ i ≤ S with r(i) > 0: σ(i) = c/r(i), σ(j) = 0, for all j 6= i. Hence the result follows. Next we consider the limit of the numerator AN of gN , as given below: AN =

hhN , 1i hwN , 1i

!N

.

(35)

Computation of the limit of AN as N → ∞ requires more information on the rate of convergence of ηN and hN . We shall use the following result from Stewart [Ste73, Thm 6.4.2] to obtain the needed rates of convergence. 19

Theorem 10 (Stewart) Suppose ν0 is a simple eigenvalue of the matrix A, with associated left eigenvector φ0 . Let E N = O(1/N ) denote a matrix with entries EN (i, j) = O(1/N ) for all i, j. Then, for some eigen-pair (eigenvalue-eigenvector pair) (νN , φN ) of the matrix A + EN νN = ν0 + O(1/N ) and φN = φ0 + O(1/N ) . Theorem 11 Let η and h be as defined in Theorem 9. Then 1 ηN = η + O N 



and 1 hN = h + O N 

Proof.



(36)

To begin, assume that r(i) 6= 0 for all i, so that R is invertible. (The result holds

even if r(i) = 0 for some i, but becomes notationally complex.) From Equation (30), we see that (η, (h, 1)) is an eigen-pair of the matrix 

 Q − µI

A=

−λ

λa





µ1   −R−1 

0

0  1/c

 .

Similarly, from Equation (29), we see that (ηN , (hN , 1)) is an eigen-pair of the matrix 

 

Q − µI µ1   CN =    λa −λ 

 Q − µI

= A+

λa

c I N

−R 0



µ1  −λ

20

 BN

−1



0  1/c



where B N = [bij ] is a diagonal matrix with bij =

  

c/r(i) c−N r(i)

for i = j ∈ {1, . . . , S}

0

otherwise

 Q − µI

µ1 

 

.

Since B N = O(1/N ), we have 

EN ≡ 



−λ

λa

 B N = O(1/N ) .

Application of Theorem 10 now gives the desired rate of convergence. Using the above Theorem, we compute the limit of AN in the next Lemma. Lemma 3 Let h be as in Theorem 9 and w = wN be as given in Equation (31). Then lim AN = ehh − w,1i .

N →∞

Proof.

Substituting Equation (36) gives "

hhN , 1i hwN , 1i

#N

"

=

hh + O(1/N ), 1i + N hw, 1i + N 



=



1 hh, 1i + o N1 N 1 hw, 1i + 1 N

+1

#N

N 

.

(37)

Taking limits in the numerator and denominator of Equation (37) shows that the right hand side converges to ehh,1i−hw,1i as N → ∞. This completes the proof. Finally, we have the following result for the limit of gN . Theorem 12 Let

  !  w(i) c/r(i)  . D = max   i:r(i)>0 h(i)

Then g ≡ lim gN ≤ Dehh − w,1i N →∞

21

Proof.

Follows from Lemmas 2 and 3, and using the fact that for positive numbers

x1 , x2 , ..., xN 1/ min{x1 , x2 , ..., xN } = max{1/x1 , 1/x2 , ..., 1/xN }.

Theorems 11 and 12 give the following theorem, which is the main result of this paper. Theorem 13 For a TS system, P (X > B) ≤ ge−ηB for g as shown in Lemma 12 and η as shown in Lemma 11. We illustrate the above results with a simple analytical example below. Example: Constant Bit Rate (CBR) Source. A CBR source is characterized by Q = [0], R = [r], and a = [1]. Duffield and Daley [DD] derive an upper bound on the probability of buffer content exceeding a given level, B, but their proof is incomplete. We achieve the same results using Theorem 13. For this case, η is the smallest positive solution to 

 µ − ηr

0 = det 

−λ

−µ λ + ηc

  

= −η[(rλ − µc) + crη] which gives η = µ/r − λ/c, as indicated by Duffield and Daley. The stability condition cµ/(rλ) > 1, which ensures that η > 0. Solving Equation (31) yields w = λ/µ, and Equation (28) gives h = c/r. Theorem 12 gives g ≤

rλ cµ

!c/r

22

ec/r−λ/µ .

Hence, Theorem 13 gives: P (X > B) ≤

rλ cµ

!c/r

ec/r−λ/µ e−(µ/r−λ/c)B ,

B ≥ 0.

In the next section we shall show how the TS bounds can be applied to the twin problems of admission control and capacity design.

6

Admission Control

Consider a TS system with parameters {a, Q, R, µ, c, λ}. The problem of admission control is to selectively admit or reject the arriving sources into the system so that the steady-state overflow probability P (X > B) is bounded above by a given  > 0, called the quality of service parameter. To do this we suppose that all parameters, except λ, are fixed and then identify the set of values of λ so that P (X > B) ≤ . Now, it is obvious (and easily shown by using coupling arguments) that P (X > B) in a TS system is an increasing function of λ. Thus, there is a maximum value of λ for which P (X > B) ≤  is satisfied. Since it is difficult to compute P (X > B) exactly for a TS system, we use the upper bound for it as given in Theorem 13. However, it is not clear that this upper bound is an increasing function of λ. For example, in the CBR example of the previous section, the upper bound is analytically available, and can be seen to be an increasing function of λ in the region of stability. In general, it is easy to show that w is an increasing function of λ. However, the behavior of h and η is harder to analyze. All our numerical experimentation, some of which is reported in Section 8, shows that the upper bound of Theorem 13 is indeed an increasing function of λ as long as the system is stable. Thus, numerically we compute λ∗ as the solution to ge−ηB =  23

(38)

where g as given in Lemma 12 and η as given in Lemma 11. Let this maximum value of λ be denoted by λ∗ . Using this value of λ∗ , we can implement the following admission control mechanism, which we shall call the TS (for Transient Sources) mechanism for admission control: If λ ≤ λ∗ , admit all the incoming sources, else admit an incoming source with probability λ∗ /λ. The definition of λ∗ guarantees that the steady-state overflow probability under this congestion control mechanism is bounded above by . The TS congestion control mechanism is described by a single parameter: λ∗ . Next we compare the performance of this congestion control mechanism to the effective bandwidth (EB) control mechanism, which we briefly describe here for completeness. The EB mechanism computes the effective bandwidth eb of a single source as the the largest B positive eigenvalue of R − log() Q. Then it allows an incoming source to enter as long as the

sum of the effective bandwidths of all the sources in the system is less than c. How do we compare the TS mechanism to the EB mechanism? First, the EB mechanism is easier to implement because it involves fewer calculations. However, the EB mechanism is based on the assumption that the sources stay in the system permanently. Hence it ignores the parameters λ and µ. Thus, its assurance of the bound on overflow probability can be valid only in the limiting case as µ → 0.

For numerical comparison we use two measures of performance: 1. The steady-state number of sources in the system, L(T S) and L(EB), 2. The steady state probability that an incoming source is blocked (and rejected), B(T S) and B(EB). We discuss these two measures below. Under the TS mechanism, the sources enter the

24

system according to a Poisson process with rate min(λ, λ∗ ). Hence, the number of sources in the system is an M/M/∞ process with arrival rate min(λ, λ∗ ) and service rate µ. Hence, L(T S) =

min(λ, λ∗ ) . µ

Under the EB mechanism, an arriving source is blocked if there are K = bc/ebc sources already in the system. Thus, the number of sources in the system is the queue length process of a M/M/K/K/K system with arrival rate λ and service rate µ. Hence PK

i(λ/µ)i /i! . i i=0 (λ/µ) /i!

L(EB) = Pi=0 K Clearly, L(T S) ≥ L(EB),

for λ < λ∗ .

For λ ≥ λ∗ , L(T S) = λ∗ /µ; and as λ → ∞, L(EB) → K. Next we study λ∗ /µ as a function of µ. As µ becomes much larger than the environment CTMC transition rates, the sources tend to stay in their initial (entering) state until they leave the system. Thus, the effective mean input rate, r¯, is driven more and more by the initial distribution, a, as µ gets larger. In the limit, as µ → ∞, we get that each source behaves as a CBR source with input rate ha, ri. Thus, the expected number of sources in the system is given by lim µ→∞

λ∗ c → . µ ha, ri

(39)

Next consider the region µ → 0. In this case, sources behave more and more like permanent sources (staying longer as µ → 0), so that the steady-state behavior of the environment process strongly influences the behavior of the system. Since the long-run number of sources in the system, L, has a Poisson distribution with mean λ∗ /µ, we expect that the limiting value of λ∗ /µ would be such that P (L > bc/¯ rc) = . 25

(40)

Next consider the probability of blocking. Under the T S mechanism we have: B(T S) = max(0, 1 − λ∗ /λ). Under the EB mechanism we get 

K

B(EB) = (λ/µ) /K!

K  X

! i

(λ/µ) /i! .

i=0

We compare these two performance measures numerically in the Section 8 for various cases.

7

Capacity Design

As in the previous section, consider a TS system with parameters {a, Q, R, µ, c, λ}. The problem of capacity design is find the smallest value of the buffer output capacity c such that the steady-state overflow probability P (X > B) is bounded above by a given  > 0. Following the same arguments as in the previous section, this reduces to solving Equation (38) for c. Taking natural logarithms on both sides of the equation, and rearranging, we get η=−

ln() ln(g) + . B B

Now consider the asymptotic region { → 0, B → ∞ :

ln() B

→ ζ}. In this asymptotic

region, η → −ζ. From Equation (27) it follows that the value of c must satisfy 

det  

µI + ζR − Q −λa



−µ1   = 0, λ − ζc

or, dividing the last row by λ, 

 µI + ζR − Q

det 



−µ1  λ − ζ λc

−a 26

 = 0.

It is clear that there is a unique solution c∗ = c/λ to the above equation. This is called the effective bandwidth of the TS system. The above discussion implies that the quality of service is guaranteed to be satisfied if c > c∗ λ, i.e., we need a capacity of c∗ per unit arrival rate. This linearity with respect to λ in a TS system is a reflection of the additivity of effective bandwidths in the permanent source systems. We illustrate the results of this section numerically in the next section.

8

Numerical Experiments

Using MATLAB routines, we investigated the behavior of several hypothetical systems (cases). In each case we use a buffer size of B = 10, with buffer service rate c = 2.5, and max overflow probability QOS criterion  = .00001. Table 1 gives the source parameters for each case. Case 1 models CBR sources. Case 2 consists of two-state “on-off” sources, while case 3 uses two-state “slow-fast” sources. Cases 4, 5, and 6 all model three-state “offslow-fast” sources, just differing in the transition rate of the CTMC environment process. Consider the case µ = 3 and λ = 5. For Case 4, the environment changes state much more slowly than sources arrive or depart. In Case 5, the transition rates for the environment CTMC and the arrival/departure process are similar. Case 6 is the more usual case of fast environment transitions relative to the arrival/departure rate. For each case, we vary µ and find λ∗ , the maximal allowable arrival rate, so that the TS bound is equal to the QOS criterion . Figure 1 shows λ∗ as a function of µ for Case 1. As expected from Equation (39), λ∗ grows linearly as µ gets large, with slope approximately λ∗ /µ. Figures 2 and 3 show λ∗ /µ as a function of µ, emphasizing the asymptotic behavior as µ → ∞. The asymptotic value, given by Equation (39), is shown as the dash-dot line in 27

Table 1: Source Characteristics, By Case Case

Type

a

R

Q

1

CBR

[1]

[1]

[0]



2

Off/On

[2/3, 1/3]



0 1



3

Slow/Fast















0 2



 



4

Off/Slow/Fast [1/3, 1/2, 1/6]

 0 0 0       0 1 0     

0 0 2





5

Off/Slow/Fast [1/3, 1/2, 1/6]

−30

30

30



−30



 −.20 .10    .225 −.30  

.16

.24



28



.075   

−.40

1.6



 

2.4

1.0  

0.75   

−4.0

 

 0 0 0   −20. 10.         Off/Slow/Fast [1/3, 1/2, 1/6]   0 1 0   22.5 −30.

0 0 2

.10  











 −2.0 1.0    2.25 −3.0  





20 

 0 0 0       0 1 0     

0 0 2

6

20 

 −20

 1 0 

[2/3, 1/3]



 −20

 0 0 

16.

24.

10.   7.5   

−40.



Figure 1: Linear Tail of Maximum Serviceable Arrival Rate: Case 1

each graph.

The horizontal dashed line in each graph shows L(EB) = K, the maximum

number of sources that would be allowed by the standard EB method. Figures 4 and 5 show λ∗ /µ as a function of µ, emphasizing the asymptotic behavior as µ → 0. The asymptotic value, imputed by Equation (40), is shown as the dashed line in each graph. Next, we compare the probabilities of blocking B(T S) and B(EB). Figures 6 and 7 plot these probabilities versus µ for cases 1 and 6, assuming a true arrival rate of λ = 5. These plots show that the EB control performs better when the expected lifetime (1/µ) is much longer than the expected interarrival time (1/λ). This is as expected.

29

Figure 2: L(T S) and L(EB) for λ >> λ∗ : Case 2

Figure 3: L(T S) and L(EB) for λ >> λ∗ : Case 6

30

Figure 4: L(T S) and L(EB) for λ >> λ∗ : Case 2

Figure 5: L(T S) and L(EB) for λ >> λ∗ : Case 6

31

Figure 6: EBW vs TS Control – Case 1

Figure 7: EBW vs TS Control – Case 6

32

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34