WO2003007622A2 - Preventive and peactive congestion control in geosynchronous satellite-switch atm networks - Google Patents

Abstract

A congestion control system and method for a satellite-switched ATM network focusing on an analysis approach for evaluating cell loss probability in the case of non-real time services through a geosynchronous satellite ATM switch is provided. The method obtains accurate performance bounds in a relatively simple setting which is still complex enough to allow a view toward generalizations of the technique.

Description

CROSS REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Patent Application No. 60/304,798 entitled "Preventive and Reactive Congestion Control in Geosynchronous Satellite- Switched ATM Networks: Sketch of an Analysis Methodology", filed July 13, 2001, which is hereby incorporated by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to preventive and reactive congestion control in geosynchronous satellite-switched ATM networks and, more particularly, to a system and method for congestion control for evaluating cell loss probability in the case of non-real time services through a geosynchronous satellite ATM switch.

2. Description of Related Art

The evolution of communication system technology since the early 1960's, when packet-switching was invented for military applications, has involved the emergence of a wide variety of techniques and technologies not envisioned even by many of the pioneers. During the same period, communication satellite technology evolved very rapidly. These technologies are now being combined to address an emerging need for quickly-installed, configurable, bandwidth-on-demand platforms and access devices to interconnect a geographically dispersed consumer and business enterprise market base.

ATM systems have generally been used in terrestrial systems for voice communications. However, the problem of signal congestion frequently occurs in ATM systems when a large number of subscribers are transmitting ATM data simultaneously. SUMMARY OF THE INVENTION

In order to overcome the problems of signal congestion in ATM switched systems, the invention provides a preventive and reactive congestion control characterizing the effect of feedback congestion control on cell loss probability in long-delay ATM networks, such as geosysnchronous satellite-switched networks. The invention also provides a preventive and reactive congestion control comparing cell loss probabilities between open-loop, preventive congestion control schemes and reactive, feedback schemes.

In accordance with these features, the invention provides a preventive and reactive congestion control which obtains accurate estimates to cell performance in the case of non-real time services without considering the issue of delay. Accurate approximation techniques are employed to obtain simple bounds which can serve as the starting point for detailed design analysis. The invention provides a method of bounding cell loss performance in the preventive case using Markov chains. In the case of feedback congestion control, the cell loss performance is bounded by using the most elementary techniques of large deviation theory. These bounds follow from the fundamental and simple techniques of Chernoff and Cramer.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is described in relation to the following drawings, in which like reference symbols refer to like elements, and wherein:

Fig. 1 shows a buffering model for reactive congestion control with feedback in accordance with the present invention;

Fig. 2 shows the application of cell loss probability bounds for a preventive mode of congestion control in accordance with an embodiment of the invention; and Fig. 3 shows a B-H level required to maintain a buffer overflow probability in accordance with an embodiment of the invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

This application incorporates herein by reference co-pending and commonly assigned U.S. patent application no. 09/975,936, entitled, "A Distributed IP Over ATM

Architecture," filed on October 15, 2001, and co-pending and commonly assigned U.S. patent application no. 09/976,271, entitled, "Distributed JP Over ATM Architecture," filed on October 15, 2001.

Reference will now be made in detail to embodiments of the invention, examples of which are illustrated in the accompanying drawings.

As described above, the system for congestion control in accordance with the invention provides an evaluation of cell loss probability in the case of non-real time services through a geosynchronous satellite ATM switch. As shown in Fig. 1, data transmitted from various terminals 30 is first sent to a satellite (not shown) and is stored to an output buffer 20 for handling before being set out to its destination. The control system in accordance with the invention first decides a discrete cell model for buffering in which time is slotted with a slot duration equal to the ATM cell duration, as shown in Fig. 1. Multiple cell durations form a

service period μ. When the buffer 20 content equals or is larger than H 35, which represents a

high level, the system send a stop-transmission message to all temώials 30 requesting that the terminals 30 to stop sending messages. When the buffer 20 content reaches L 40, a low level, the system sends a continue-transmission message to all terminals 30 requesting that the terminals 30 starting sending messages. In certain situations, there is a delay d from the time stop sent to the terminals to the time when cells stop arriving at buffer 20. It can be assumed that cell arrivals are independent and Bernoulli since near adjacent cells will typically be from different sources (i.e., user terminals), especially if the data rates are low. The extension to burst scale modeling requires a different of analytical tools.

As an example, assume that up to N cells can arrive during a service duration μ,

so that the maximum cell rate into the switch is N/μ, and that cell arrivals are independent Bernoulli random variables with probability of cell arrival p. Then the number of cell arrivals, η(k), during service intervals k = 1,2,... are independent binomial random variables with parmeters N and p. We define

η_n = Prfø (k) = = 0,1,2, K, N (0a)

Let Xk, k = 0,1,2,... be the buffer content in cells at the beginning of service interval k. The in-service cell is part of the buffer content. The cell that was just serviced in interval k-1 is not included in the buffer content. It follows that the buffer content evolves according to

χ_k = πάΩ. {max{ _M - 1,0} + η(k), B}, k = 1,2, K (la) We may assume that X₀ takes any convenient value, which we define below.

A terminal's response to a stop-transmission message is to stop transmitting cells until it receives a continue-transmission message; the former is triggered when the buffer content meets or exceeds H 35 in Figure 1, and the latter is triggered when the buffer content is at L 40. The exact model of buffer content after it meets or exceeds H 35 is as follows: After a finite delay, d, no cells are received into the buffer. The buffer is then emptied of the buffered cells at the constant service rate 1/μ until level L 40 is reached. Note that d is the amount of time it takes to notify the terminals of a level hit plus the amount of time needed for the cells that are in-route to the satellite to enter the buffer.

Upon receipt of a continue-transmission message, a terminal immediately begins transmitting cells. In this case, the buffer 20 does not receive any cells until Δ= d/μ service periods after L 35 is reached. At this time the buffer 20 content is max {L - Δ , 0} cells. Since Δ is large, the buffer 20 content may be 0. During the next service interval, cells can enter the buffer 20. Then at a number of service intervals later, as governed by the Bernoulli arrival process, the buffer 20 content is >H.

In the case of feedback congestion control, the only period of interest from a loss probability point-of-view is from the time of entrance into the region of buffer content >H and back out again. Under appropriate conditions on B - H, during this interval cells can be lost due to buffer overflow. Statistics of this passage and approximations involving an infinite capacity queue are developed. For example, Fig. 2 shows the instance where the probability of buffer overflow exponent for N=2. Fig. 3 shows a B-H required to maintain the buffer probability in accordance with the invention.

Define ki to be the first in a sequence of integers such that ki >1, X^-i <H, but

X_kj ≥H. For j=2,3,..., let k_j =min{k : k > k_j.i, X_k-i, <H,Xk >H}. By construction, the sequence

kj, k₂,... exists and is monotonically increasing. The X . does not exceed B, but, for each j, each

of X_kj+m,.m = 1, 2,K, Δ can become equal to B probabilistically during the delay interval of duration d = Δ μ between the time that H is met or exceeded and the time that the last cell is received from terminals. The waiting times to level L, measured in service intervals, which

occur during terminal off-periods as the cells are emptied from the buffer, are equal to Xkj₊Δ- L; these can be equal to B-L in the case of an overflow. Consider the 2-dimensional random variables (kj, Xkj), j =1, 2, K. Their joint distribution is independent of j since each ascent from L - Δ to or past H is statistically identical. We therefore deal with (ki , X_k ) only. The random

variable k, is the first passage time into the set of integers {x : x > H}.

The buffer content {X_k k = 0,1,2, A } is a Markov chain with a finite state space,

but the transition matrices depend on service interval, k, when X_kl ≥ H. We avoid this

complication by analyzing an infinite capacity queue: Define {x : k - 0,1,2, Λ}to be the buffer

content process for an infinite capacity queue with the same arrival process (0a) as considered for the finite capacity buffer.

In the case where no feedback information on L/H level crossings is given, the terminals transmit under constraints that maintain cell loss performance. The constraints are typically relayed to the terminals via signaling. This mode of congestion control is called preventive. Various methodologies can be applied in this case in addition to connection admission control. The desired result of these approaches is to maintain a state of equilibrium of the buffer content process such that desired cell loss performance is maintained. In general, this requires that Np <1. A mathematical model of this mode of operation is described in greater detail below:

Define P c _L ( k ) to be the cell loss probability process for the finite buffer content process { X k : k = 0, 1, 2, Λ} . From (la):

X " =

- l,θ}+ η(k), k -l,2,K (lb) The buffer content {X" : k = 0,1,2, Λ} is a homogeneous Markov chain with an

infinite state space. Without exchange of feedback information on L/H level crossings as in the reactive congestion control regime, one can assume that the buffer content process extends backward in time indefinitely. It follows that the Markov chain is governed by its stationary distribution.

These concepts may be summarized in the hypothesis of the following lemma.

Lemma 1: Let { " : k = 0,1,2, Δ) be the content process for the infinite capacity

queue with the same arrival process as the original finite buffer content process

{X^™ : k = 0,1,2, A}, and assume that it is stationary. Then P _{C L} ( k ) ⁼ P _{C L} is defined

independent of service interval, k, and

P_CL ≤ -^ ϊ^>v{x^∞ _k > B fo k = 1,2, K (2)

Np

Next determine the stationary distribution of the process {X^™ : k = 0,1,2, A}.

Lemma 2: The transition matrix for the evolution equation (lb) is

where

Pn =

The block matrix, Z, is the all zero matrix. It can be shown by verifying qP = q that the unique stationary distribution, q = (q₀, q_ls q, K ) , is given by

\N N q° = (ι-p)^N-p p<l,N>2) (3)

Proof: The conditions ^ q = 1 and qP = q can be shown directly. n=\

Thus, q is a stationary probability distribution. A nonzero transition probability exists between any two buffer content states. This statement is what is meant by irreducible and positive recurrent. Therefore, applying a well-known theorem, a unique stationary distribution with positive elements exists. This completes the proof.

In the case N=l and B>1, no overflow is possible when Np<l. This is clear since the peak cell rate and service rate are identical. In this case, the stationary distribution used in place of (3) is q₀ =1- p , qi, = p , and q_n = 0 for n > 2, as expected.

Theorem 1: If the content process χ~ : k = 0,l,2,Λ for the infinite capacity

queue with the same arrival process as the original finite buffer content process

| " : k = 0,1₅2,ΛJ is governed by its stationary distribution (3), then the cell loss probability,

PCL, is bounded as follows:

Proof: Using Lemma 1 and the stationary distribution (3), the stationary cell loss probability is bounded such that

NB

Np ^{{Λk j} N _nέ ⁿ Np 1- p

The final equality is obtained using the formula for the sum of a geometric series. This completes the proof When the feedback control model formulated above is considered, the possibility of cell loss will only occur during the passage times through {x : x > H} . As discussed below,

the cell loss probability is bounded.

The use of feedback information opens the possibility of operating in the heavy traffic region Np >-l, which is ruled out in the preventive case. To bound the cell loss probability process in this case, a restriction on B-H is required. This can be made clear in a manner similar to the familiar analysis of an M/M l queue. Using the strong law of large

numbers applied to X" _+m = X^™ + ∑ (η_n - 1), m = 1 ,2, K Δ, it is concluded that n=l

(X_k1+m ^~ Xi /^{m >} NP - 1 J as Hi > ∞ , which applies also when m > Δ if Δ i

sufficiently large. Thus (x _ϋ"_1+m - X γ "_q )_/~ m -_^(N _^p - 1 -) / as m increases.

When Np > 1, the probability of overflow is controlled by the size of B-H. The next theorem gives a sufficient condition insuring a well-behaved overflow probability and provides a bound to the cell loss probability process for a finite buffer.

m

Theorem 2: Define (B-H)_min = m(Np-l) + (N-l) and S_m = ∑ΗCk-i + i). If

1=1

B-H=(B-H)min + β, β > 0, then

NpP_0L(k_{1 +}m)≤Prfc_1+m > B}<Pr{S_m ≥ β _{+ m}Np}< e-**^> (41) where y miN — y α(y) = y log — + (mN - y) log mN log mN (42) p 1 - p and (rnNp) = 0. Proof: Since

X _m + m = H + (χ;-H)₊S_m<H ₊ N-l + S_m, we have

NpP k, +m)<Prfc_l+m>B}<Pr{S_m >x + m}, ^{( 3)}

by Lemma 1 , where x = B-H-N + 1.

Given a random variable Y with moment generating function, φ() = Ele L

we have

Pr{Y > y}= J _!xx≥y}dF_γ(x) < J _{__y}e^^y)dF_Y(x)

= E[e^t(Y-^y)l_{γ≥y}]<E[e^t(Y-^y)]=e-^{ty-'^ogP(t)}

The right-hand side will be minimized when the exponent {ty — lθgζ£>(t)}is

maximized. It is shown using standard convexity arguments in, that

&{y) = SUpjty — log^>(t)| is the maximizing exponent. It is also shown there that a is

convex with unique minimum at EY and α(EY) = 0. The special case of this result for the

binomial sum S_m is the result of interest in this application.

For binomial inputs, we have:

_x ✓ _M x + m , ,_τ ., mN-x-m __. __ α(x + m) = (x + m)log (mN - x - m)log mNlogmN

P i-p

It is clear that, if X + m = E(S_m) = mNp, then α(x + m) = 0. Now assume

B-H-β=m(Np-l) + (N-l) = (B-H)_mm. Then x + m = B-H-N + l + m

= (B-H)_min+β-N + l + m

= mNp+β

This completes the proof.

The term N - 1 in the lower bound, (B-H)_mi_n, to B-H in Theorem 2 guarantees that, in all heavy load cases with Np >1, no cells are lost at the first passage k, into {x : x > H}.

While specific embodiments of the invention have been described herein, it will be apparent to those skilled in the art that various modifications may be made without departing from the spirit and scope of the invention.

Claims

CLAIMSWe claim:

1. A method for congestion control in a satellite-switched asynchronous-transfer-mode (ATM) network, in which the ATM network includes an output buffer, comprising the steps of: providing data from a ground base unit to a satellite; detecting an occurrence of congestion in the network; and configuring the output buffer to minimize the congestion.

2. A method for congestion control in a satellite-switched asynchronous-transfer-mode (ATM) network, comprising the steps of: determining a desired cell loss probability based upon a satellite capability; calculating transmission characteristics of a ground station based upon the desired cell loss probability and data characteristics of the satellite; and controlling the ground station transmission to control congestion based upon the calculation.