GB2376838A

GB2376838A - A method of evaluating global-clocked Banyan network offered lengthened messages

Info

Publication number: GB2376838A
Application number: GB0219518A
Authority: GB
Inventors: Lyaw I-Pyen
Original assignee: Individual
Current assignee: Individual
Priority date: 2002-08-21
Filing date: 2002-08-21
Publication date: 2002-12-24
Anticipated expiration: 2022-08-21
Also published as: GB2376838B; GB0219518D0

Abstract

An objective of the invention is to overcome the drawback that the banyan-networks' space availability analyzed/simulated cannot be effectively propagated. A further objective is to upgrade the network's throughput. It has been proven that a global clock is better. In accordance with the invention, a packet can still move even when its destined room is full if that destined queue can transmit simultaneously. In accordance with the method of the invention, the message-length effects can be predicted.

Description

A METHOD OF EVALUATING GLOBAL-CLOCKED BANYAN NETWORKS OFFERED LENGTHENED MESSAGES Field of the Invention This invention is based upon a paper, entitled"ANALYSIS OF BANYAN NETWORKS OFFERED TRAFFIC WITH GEOMETRICALLY DISTRIBUTED MESSAGE LENGTHS"and published in the journal"IEE Proc.-Commun., Vol. 142, No. 5 by I.-P. Lyaw and D. M. Koppelman in October 1995, and is used to refine ineffective usage of queues'room in banyans where a packet can find its destined room's space only when that room is not full. This invention is to increase the packets'movement even when the destined room is full in order to fully utilize the limited space.

Background of the Invention Banyan networks are extensively used in communications and parallel-computing systems. It is necessary to analyze the performances of such networks in order to evaluate the system designs and understand the networks themselves. Many banyan-network analysis methods have been reported. However, the bulk of those analyses were for networks offered uniform message length. Since networks used for communication switches and parallel computers must carry messages having varying lengths, a nonuniform-message-length analysis is needed.

Analysed networks will be specified by a 3-tuple (n, a, m). Such a network consists of n stages, numbered 1, the input stage, to n, the output stage. Each stage contains an-'a-input, a-output switching elements (SEs). Each switching element consists of a m-slot queues, each connected to the crossbar inputs. Links connect SEs in adjacent stages and first-and last-stage SEs to network inputs and outputs, respectively.

During a cycle, a network input can either be idle, have a head packet arriving, or have a nonhead packet arriving. A three-state discrete-time Markov chain with states labeled I (idle), S (start), and A (active) is used to describe message arrival times and lengths. Let x # y denote a transition from state x to y and let THy denote the corresponding transition probability. An input not receiving a packet during a cycle is

modeled by transitions x I, x c= II, S, Al. A head packet arrival is modeled by transitions x S, x c II, S, Al ; a message continuing is modeled by transitions x- A, x E {5, A}.

A network is modeled by n Markov-chain pairs, each of which characterizes a stage. One pair member, named the queue model, characterizes a switching element queue. The other pair member is named the head-of-line (HOL) model, characterizing a switching-element HOL system. All queue and HOL models making up a network are statistically independent of each other. The state distributions are solved by iteratively computing state distributions and transition probabilities.

Each stage is modeled by two state machines. One is the head-of-line (HOL) model, and the other is the queue model. The queue model has two sets of states: one set is for queues into which a message is entering, and the other set of states is for queues into which no message is entering. Transitions for the HOL model are based, among other things, on the probability that a message using a switching-element output will end.

Transitions are also based on the probability that there will be space in the next-stage queue given the type of packet, head or nonhead. These space probabilities are determined from the queue model.

The HOL model is needed because messages are transmitted in several consecutive packets which cannot be interrupted. If the correlation among packets by prior contention, captured in the HOL model, is instead neglected, predictions will be inaccurate, especially for large message lengths.

In the previous work, queue-model transition probabilities are a function of arrival rate, service rate, and expected message length. Let , 0 < / ? K, denote the arrival rate, the probability a new message will be ready to move into a stage-j queue given that the queue is in state !,- or ! . Four distinct values of arrival rate y, are computed per queue : rO, for an empty queue, r,, j = ru, where 0 < i < m -1, for a queue with 1 to m - 2 packets, 3 rm for a queue with one slot free, and rm, for a full queue.

The stationary probabilities must satisfy a restricted Markov-chain in which a packet in a queue head can never move into a next-stage queue if that destined queue is full.

Stage-j space probabilities are computed so the flow rate leaving a stage-j HOL slot is equal to the flow rate entering a stage- (j+l) queue. The probability that a packet ready to enter a next-stage queue is successful to enter is limited by the condition that there must be at least one slot free in that destined queue.

The flow rate is determined by the arriving traffic, but also by the fraction of time that traffic is not blocked that leads to the insufficiency of flow rate.

The delay of a message is defined to be the number of cycles that the head packet is in the network. The normalized delay is defined as the delay divided by the number of stages. The waiting time of a message in a queue is defined to be the number of cycles that the head packet spends in the queue. The delay then is the sum of the waiting times. The total time a message spends in the network is the delay plus the message length minus

one. The expected waiting time is found by first computing, for 0 i < m, the waiting time, w,,,, for a head packet arriving at a stage-j queue that had i packets in the previous cycle. Let s'be the service probability of a HOL packet that is a head packet. The expected waiting time of a head packet at a stage- ;' queue HOL slot is then 1/s'j. The expected waiting time of a HOL packet of unknown type is 1/s}. If the queue had i

packets in the cycle before a message arrived then w,,, = il i i The expected waiting time w, can be induced and the normalized delay is then E, /n.

In the prior art, one main disadvantage is that a packet in a queue head can never move into a next-stage's queue if that destined queue is full.

Summary of the Invention An objective of the invention is to overcome the drawback that the banyans'space availability analyzed/simulated cannot be effectively propagated. A further objective is to upgrade the network's throughput. It has been proven that a global clock is better. In accordance with the invention, a packet can still move even when its destined room is full if that destined queue can transmit simultaneously. In accordance with the method of the invention, the message-length effects can be predicted.

One main shortcoming of the prior art is that a packet in a queue head can never move into a next-stage queue if that destined queue is full.

On the contrary, a packet in accordance with the invention can enter a next-stage's full queue if that queue can transmit a packet forward during the same cycle. An example is shown in Figure 1. In accordance with the invention, only three distinct values of arrival rate are needed: ro for an empty queue,

= where 0 < i < m, for a queue with 1 to m -1 packets, and ? for a full queue.

Brief Description of the Drawings Figure 1 illustrates an example of a packet entering a full queue that is transmitting an old packet forward simultaneously; Figure 2 illustrates throughput against message length for (5,2, 8) and (5,4, 8) networks, #=1 ;

Figure 3 illustrates delay against message length for (5, 2, 8) and (5, 4, 8) networks, !, = ! ; Figure 4 illustrates delay against arrival rate in (5, 2, 8) networks for message length 8; and Figure 5 illustrates delay against queue size in (5,2, m) networks for #=1 and message length 8.

Detailed Description of the Invention The network consists of n stages, numbered 1, the input stage, to n, the output stage. Each stage contains an-l a-input, a-output switching element. Each switching element consists of a m-slot queues, each connected to a crossbar input. Links connect switching elements in adjacent stages and first-and last-stage switching elements to network inputs and outputs, respectively. The links can be connected in any pattern for which there is exactly one path between all network input/output pairs.

The parameters appointed in this invention are: ! : the queue-model states where 0 ; c < denotes the number of packets in the queue and y E {T, T, #} denotes the packet type in the last occupied slot of a queue. The symbol y = T if the last occupied

slot holds the tail packet of a message, y = T if the last occupied slot holds a nontail packet of a message, and y = 0 if x = 0; pj(Ix,y) : the probability that stage-j queue is instate lx, y ; si the service rate, i. e. , the probability that a stage-j HOL packet is able to move forward; s'j : the service rate of a head packet, i. e. , the probability that a stagej HOL packet is able to move forward, given that the packet is a head packet; r@,j : the arrival rate, the probability a new message will be ready to move into a stage-j queue given that the queue is in state !or)g, ; VE : the empty-queue arrival rate in stage j ; #H,J : the probability that a head packet in the HOL slot of an active stage-j queue will find space in its next-stage queue; PH : the probability that a nonhead packet in the HOL slot of an active stage-j queue will find space in its next-stage queue; wt,j : the waiting time for a head packet arriving at a stage-j queue that had i packets in the previous cycle; wu : the expected waiting time for a head packet arriving at a stagej queue; A : the probability of a new-message's head-packet arrival given the queue into which no earlier message is still entering; p the flow rate of the network; : the probability that an entering packet is a tail packet of a message;

m : the queue size; and n : the number of stages the network consists of.

Analysis : In accordance with the invention, the calculation of the previous work's rm,@ is first modified to the following equation, given 2 < j < n,

By using the modified a set of equations shorter than those in the previous work for the stationary probabilities of queue-model states is expressed as follows:

for 2/-1 and l < yK.

The previous work's expression for the probability that a head packet successfully enters a stage- (j+ 1) queue is then refined to the expression below:

Given 1#j#n-1, and the head-packet space availability becomes the following equation accordingly,

Under the condition of 1 < < -1, we can derive the following nonhead-packet space availability by using similar reasoning,

The values of head-packet space availability and nonhead-packet space availability in accordance with the invention are larger than those in the previous work.

The flow rate of the network is hence upgraded to

so as to fit in with concern perfectly.

Computation of Delay : The waiting time for a head packet arriving

at a stage- ;' queue that had i packets in the previous cycle should change from the previous work's to

I 1/, if i =0 w=/+1/-1, if0 < < (/-l)/+l/, if/=

so that the expected waiting time should be

Results : The prediction of message-length effects on global-clocked banyans offered saturating traffic can be seen in Figures 2 and 3. The effect of message size is still clearly modeled. As with other analyses of this type, the throughput is overestimated. The delay in simulated and analyzed systems still closely match.

The delay at varying arrival rates and the effect of queue size are also predicted as can be seen in Figures 4 and 5. They match simulations closely, too, especially for smaller queue sizes.

Conclusions : The analysis was tested against simulations. The results show that message-length effects for global-clocked banyan networks are still effectively modeled. In particular, the global clock can make the space availability be effectively propagated through the entire machine so that the higher throughput than the previous work's is generated.

Claims

CLAIMS 1. A method of evaluating global-clocked banyan networks offered lengthened messages, wherein each of said networks consists of n stages, numbered 1, the input stage, to n, the output stage, each stage containing an-I a-input and a-output switching elements (SEs), each switching element consisting of a m-slot queues, each queue connected to a crossbar input; and wherein links connect SEs in adjacent stages and first-and last-stage SEs to network inputs and outputs, respectively, the links can be connected in any pattern for which there is exactly one path between all network input/output pairs, and the definitions of parameters are:

for the queue-model states where 0 x m denotes the number of packets in the queue and y E {T, T, 0} denotes the packet type in the last occupied slot of a queue ; the symbol y = T if the last occupied slot holds the tail packet of a message, y = T if the last occupied slot holds a nontail packet of a message, and y = # if x = 0 ; PJ (I. t, y) for the probability that stage-j queue is in state Ix,y ; si for the service rate, i. e. , the probability that a stage-j HOL packet is able to move forward; r,, for the arrival rate, the probability a new message will be ready to move into a stage-j queue given that the queue is in state ", T or Io, ; for the flow rate of the networks; //for the probability that an entering packet is a tail packet of a message;

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m for the queue size; and n for the number of stages the network consists of ; said method being characterized in that a packet can still enter a next-stage's full queue if that queue can transmit a packet forward during the same cycle, ascribed to the global clock that makes the space availability be effectively propagated through the entire network, in which three distinct values of arrival rate are needed, namely, ro, j for an empty queue,

rN, j where 0 < i < m, for a queue with 1 to m -1 packets, and ? rm. for a full queue, wherein the judgment on ru. j is

for 2 s j S n, whereby a certain equation set used for the stationary probabilities of queue-model states is generated :

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for 2 S ism -1 and 1#j#n.
2. A method according to claim 1, characterized in that the head-packet space availability is

and the nonhead-packet space availability is

for 1'-1,wherein/and/ are the probabilities that a head packet and a nonhead packet, respectively, in the head-of-line slot of an active stage-j queue will find space in their next-stage queues, and wherein #E,J+1 is the empty-queue arrival rate in stage j+1.
3. A method according to claim 1, characterized in that the flow rate of the network is

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A is the probability of a new-message's head-packet arrival given the queue into which no earlier message is still entering.