Switching Theory: Architecture and Performance in Broadband ATM Networks phần 10 ppsx

Overall switch performance After analyzing the loss performance occurring in the interconnection network due to thefinite number of network stages, attention is now given to the other tw

Performance Evaluation and Comparison 379

SEs the Shuffleout switch gives the best performance owing to its shortest path routing which

is more effective than the simple bit self-routing applied in the Shuffle Self-Routing andRerouting switch With the more complex SEs the Shuffleout SEs still perform the best,with the Shuffle Self-Routing giving a rather close performance especially for large networks.The Dual Shuffle switch gives a significantly higher loss probability for the same networkstages

We would like to conclude this loss performance review, by comparing the number of

stages K required for a variable network size to guarantee a loss performance of for thesix architectures based on deflection routing just considered: the results are given inFigure 9.40, respectively All the curves grow almost linearly with the logarithmic switch size.Therefore we can conclude that minimum complexity of the order of characterizesnot only the Dual Shuffle switch, as pointed out in [Lie94], but all the other architecturesbased on deflection routing It can be shown that this property applies also to the Tandem Ban-yan switch With this feature established, we need to look at the gradient and the stage numberfor a minimum size switch in order to identify the real optimum architecture According toour previous considerations, Shuffleout requires the minimum amount of hardware in theinterconnection network to attain a given loss figure, whereas the Shuffle Self-Routing (DualShuffle) needs the maximum amount of hardware with ( ) SEs The conclusions todraw from these figures are that Shuffleout gives the best cost/performance ratio at theexpense of implementing shortest-path routing If bit-by-bit routing is preferred, Shuffle Self-Routing and Rerouting with extended routing provide the best solution with and SEs Analogous results have been obtained for a different loss probability target

Figure 9.37 Loss performance comparison for different architectures

382 ATM Switching with Arbitrary-Depth Blocking Networks

9.4.7 Overall switch performance

After analyzing the loss performance occurring in the interconnection network due to thefinite number of network stages, attention is now given to the other two components wherecell loss can take place, that is the output concentrator and the output queue Recall that ingeneral concentrators are not equipped in the Tandem Banyan switch owing to the small num-ber of lines feeding each output queue

[Yeh87], we assume that all the local outlets feeding the concentrator carry the same loadand are independent of one another, so that the packet loss probability in the concentrator isgiven by

(9.16)

Figure 9.41 shows the loss performance of a concentrator with output lines for avariable offered load level and three values of the number of its inlets, , whichalso represent the number of switching stages in the interconnection network It is observedthat the performance improves as the number of sources decreases for a given offered load as alarger set of sources with lower individual load always represents a statistically worse situation

In any case the number of concentrator outlets can be easily selected so as to provide the targetloss performance once the stage number and the desired load level are known Alternatively if

and C are preassigned, a suitable maximum load level can be suitably selected according to

the performance target

Figure 9.41 Loss performance in the concentrator

Performance Evaluation and Comparison 383

in the Tandem Banyan switch which does not employ concentrators Each output

queue is fed by L links, where for all the architectures with output concentrators and

for the Tandem Banyan switch So the cell arrival process at the queue has a binomial

distribution, that is the probability of i cells received in a slot is

The server of the queue transmits one packet per slot, therefore the output queue can beclassified as discrete-time , where B o (cells) is the output queue capacity

By solving numerically this queue (see Appendix), the results in Figure 9.42 are obtained for

queue is equipped with output lines As one might intuitively expect, limiting theoffered load level is mandatory to satisfy a given loss performance target

Observe that the above evaluations of the cell loss in the concentrator and in the outputqueue are based on the assumption that all the traffic sources generate the same traffic This

is clearly an approximation since the lines feeding each concentrator give a load that decreasesfrom 1 to (the load carried by the network stages decreases due to the earlier networkexits) Nevertheless, our analysis is conservative since it gives pessimistic results compared toreality: assuming homogeneous arrivals always corresponds to the worst case compared to anytype of heterogeneous arrivals with the same load [Yeh87]

The total packet loss probability given by Equation 9.1 is plotted in Figure 9.43 for the

Figure 9.42 Loss performance in the output queue

–

Output queue size, Bo

Switch Architectures with Parallel Switching Planes 385

parallel planes is shown in Figure 9.44 where each plane includes switching blocks fore each output queue is fed now by links As with the basic architectures based ondeflection routing, one switching block consists of one switching stage with the Shuffleout,Shuffle Self-Routing and Rerouting switch, whereas it includes a banyan network with stages with the Tandem Banyan switch Therefore the SEs have now the same size as inthe respective basic architectures In the case of the Dual Shuffle switch, two planes alreadyexist in the basic version, which under the traffic sharing operation includes also the splitters.Considering the parallel architecture in the Dual Shuffle switch means that each SE in a planehas its own local outlets to the output queues Therefore once we merge the SEs in the samestage and row into a single SE, whose internal “core” structure is either crossbar or ban-yan, the basic SE of the single resulting plane now has the size

There-The performance of architectures using parallel planes is now evaluated using computersimulation The loss performance of the parallel Shuffleout switch is compared in Figure 9.45with the corresponding data for the basic architecture under maximum load It is noted thatusing two planes reduces the number of stages compared to the single plane case by a factor inthe range 10–20%; larger networks enable the largest saving For example a loss target of requires 47 stages in the basic architecture and 39 stages in the parallel one for Nevertheless, the parallel architecture is altogether more complex, since its total number ofstages (and hence the number of links to each output queue) is Similar com-ments apply using interstage bridging and extended routing, when applicable, in all theversions of the three architectures Shuffleout, Shuffle Self-Routing and Rerouting Table 9.2gives the number of stages required to guarantee a network packet loss probability smaller than

Figure 9.44 Model of ATM switch architecture with deflection routing and parallel planes

0 1

N-2 N-1

C C

C C

Interblock connection pattern Interblock connection pattern

Switching block Switching block Switching block

388 ATM Switching with Arbitrary-Depth Blocking Networks

and to the Shuffle-Self-Routing switch have been described in [Dec91b], Zar93b, respectively.The analysis of Shuffleout with SEs of generic size is described in [Bas94]

The traffic performance of deflection-based architectures has been evaluated based on therandom and uniform traffic assumption It is worth pointing out that traffic correlation has noimpact on the dimensioning of the number of stages in ATM switches with arbitrary-depthnetworks In fact the interconnection network is internally unbuffered and the only queueingtakes place at the output interfaces Therefore the correlation of traffic patterns just affects thedimensioning of the output queues of the structure This kind of engineering problem hasbeen studied for example in [Hou89] Non-uniformity in the traffic pattern, that is unbal-anced in the output side, has been studied in [Bas92] for the Shuffleout architecture Non-uniform traffic patterns in the Shuffleout switch have been studied in [Gia96]

9.7 References

[Awd94] R.Y Awdeh, H.T Mouftah, “Design and performance analysis of an output-buffering ATM

switch with complexity of O(Nlog2N)”, Proc of ICC 94, New Orleans, LA, May 1994, pp.

420-424.

[Bas92] S Bassi, M Decina, A Pattavina, “Performance analysis of the ATM Shuffleout switching

architecture under non-uniform traffic patterns”, Proc of INFOCOM 92, Florence, Italy,

π n

Number of network stages, K

b×2b

References 389

[Bas94] S Bassi, M Decina, P Giacomazzi, A Pattavina, “Multistage shuffle networks with shortest

path and deflection routing: the open-loop Shuffleout”, IEEE Trans on Commun., Vol 42,

No 10, Oct 1994, pp 2881-2889.

[Dec91a] M Decina, P Giacomazzi, A Pattavina, “Shuffle interconnection networks with deflection

routing for ATM switching: the Open-Loop Shuffleout”, Proc of 13th Int Teletraffic

Con-gress, Copenhagen, Denmark, June 1991, pp 27-34.

[Dec91b] M Decina, P Giacomazzi, A Pattavina, “Shuffle interconnection networks with deflection

routing for ATM switching: the Closed-Loop Shuffleout”, Proc of INFOCOM 91, Bal

Harbour, FL, Apr 1991, pp 1254-1263.

[Dec92] M Decina, F Masetti, A Pattavina, C Sironi, “Shuffleout architectures for ATM

switch-ing”, Proc of Int Switching Symp., Yokohama, Japan, Oct 1992, Vol 2, pp 176-180.

[Gia96] P Giacomazzi, A Pattavina, “Performance analysis of the ATM Shuffleout switch under

arbitrary non-uniform traffic patterns”, IEEE Trans on Commun., Vol 44, No.11, Nov.

1996.

[Hou89] T.-C Hou, “Buffer sizing for synchronous self-routing broadband packet switches with

bursty traffic”, Int J of Digital and Analog Commun Systems, Vol 2, Oct.-Dec 1989, pp.

253-260.

[Lie94] S.C Liew, T.T Lee, “N log N dual shuffle-exchange network with error correcting

rout-ing”, IEEE Trans on Commun., Vol 42, No 2-4, Feb.-Apr 1994, pp 754-766.

[Mas92] F Masetti, A Pattavina, C Sironi, “The ATM Shuffleout switching fabric: design and

implementation issues”, European Trans on Telecommun and Related Technol., Vol 3, No 2,

Mar.-Apr 1992, pp 157-165.

[Tob91] F.A Tobagi, T Kwok, F.M Chiussi, “Architecture, performance and implementation of the

tandem banyan fast packet switch”, IEEE J on Selected Areas in Commun., Vol 9, No 8, Oct.

1991, pp 1173-1193.

[Uru91] S Urushidani, “A high performance self-routing switch for broadband ISDN”, IEEE J on

Selected Areas in Commun., Vol 9, No 8, Oct 1991, pp 1194-1204.

[Wid91] I Widjaja, “Tandem banyan switching fabric with dilation”, Electronics Letters, Vol 27, No.

19, Sept 1991, pp.1770-1772.

[Yeh87] Y.S Yeh, M.G Hluchyj, A.S Acampora, “The knockout switch: a simple, modular

architec-ture for high-performance packet switching”, IEEE J on Selected Areas in Commun., Vol SAC-5, No 8, Oct 1987, pp 1274-1283.

[Zar93a] R Zarour, H.T Mouftah, “Bridged shuffle-exchange network: a high performance

self-routing ATM switch”, Proc of ICC 93, Geneva, CH, June 1993, pp 696-700.

[Zar93b] R Zarour, H.T Mouftah, “The closed bridged shuffle-exchange network: a high

perfor-mance self-routing ATM switch”, Proc of GLOBECOM 93, Houston, TX, Nov 1993, pp.

1164-1168.

390 ATM Switching with Arbitrary-Depth Blocking Networks

9.8 Problems

9.1 Find the routing algorithm for an ATM switch using deflection routing derived from the Rerouting switch The interconnection network of this switch is built cascading Baseline networks rather than reverse SW-banyan networks with last and first stages of two cascaded networks merged, as in Rerouting.

9.2 Derive the modification to be applied to the analytical models developed in Section 9.4 to account for the availability of multiple planes.

9.3 Use the models derived in Problem 9.2 to compute the packet loss probability of different switch architectures and verify how the analytical results match those given by computer simulation reported in Figure 9.45 and Table 9.2.

9.4 Extend the analytical model of Shuffleout to account for the availability of bridges.

9.5 Extend the analytical model of Shuffle Self-Routing to account for the availability of bridges 9.6 Verify that Equations 9.3, 9.4 and 9.6 of Shuffleout simplify into Equations 9.9 and 9.11 of Shuffle Self-Routing.

9.7 Find the minimum number of stages in the Shuffleout switch that makes null the packet loss probability in the interconnection network.

9.8 Repeat Problem 9.7 for the Rerouting switch.

9.9 Repeat Problem 9.7 for the Tandem Banyan switch.

16×16

Appendix Synchronous Queues

A characteristic common to all the queues considered here is their synchronous behavior, ing that customers join and leave a queue only at discrete epochs nt that areinteger multiples of the basic time interval t that we call a slot Furthermore, starts and ends ofservice can only occur at slot boundaries For the sake of convenience we adopt the slot dura-tion as the basic time unit by expressing all the time measures in slots, so that state transitionscan take place only at times n Unless specified otherwise the queue discipline

mean-is assumed to be first-in–first-out (FIFO)

The main random variables characterizing the queue are:

• A: arrival size, that is number of service requests offered to the queue in a given timeperiod;

• W: waiting line size, that is number of customers in the queue waiting for the server ability;

avail-• Q: queue size, that is total number of customers in the queue;

• θ: service time, that is amount of service requested to the queue by the customer;

• η: waiting time, that is time spent in the queue by a customer before its service starts;

• δ: queueing time, that is the total time spent by a customer in the queue (waiting time + vice time)

ser-In the following λ will denote the average arrival rate to the queue, which for a nous queue will be indicated by p expressing the probability that a service request is received in

synchro-a slot by the queue Two other psynchro-arsynchro-ameters specifying the behsynchro-avior of synchro-a queue other thsynchro-an themoments of the above random variables can be defined:

• ρ: server utilization factor, that is the time fraction in which each server in the queue is busy;

• π: loss probability, that is probability that a new customer is not accepted by the queue

n= 1 2, ,…

n=1 2, ,…

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Switching Theory: Architecture and Performance in Broadband ATM Networks

Achille Pattavina Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-96338-0 (Hardback); 0-470-84191-5 (Electronic)

392 Synchronous Queues

A.1 Synchronous Single-server Queues

In this section synchronous queues with one server are studied Queues with deterministic vice time are examined first, in the case of both Poisson and geometric arrival processes Thenqueues with general distribution of the service time are considered with infinite or finite wait-ing line

ser-A.1.1 The M / D /1 queue

We are interested here in the evaluation of the synchronous queue in which the tomer arrivals occur only at discrete time epochs and the arrival process is Poisson We aregoing to show that this queueing system has a close relationship with the (asynchronous)

in which arrivals and starts of service can take place at any epoch of a continuoustime axis We assume that a queue is asynchronous if not specified otherwise Since the study

of the queue relies on the analysis of an queue (see, e.g [Gro85]), we willbriefly examine this latter queue

A.1.1.1 The asynchronous M/G/1 queue

In an queue the customers join the queue according to a Poisson process with age arrival rate λ and the service times are independent and identically distributed (IID) with

aver-an arbitrary probability distribution Although in general such a queue is not a Markov process,

it is possible to identify a set of “renewal” epochs in which the Markov property of the systemholds Through this technique an imbedded Markov chain can be identified that enables us tostudy the properties characterizing the system at these epochs These renewal epochs are thetimes at which customers complete the service in the queue and thus leave the system Thenumber of customers left behind by the n-th departing customer who leaves the system at

(A.1)

in which is the number of arrivals during the service time of the n-th customer

The evolution of the system can be described by defining:

• the distribution of arrivals in a service time

We assume that a steady state is achievable (the necessary conditions are specified in anyqueueing theory book, see e.g [Gro85]) and thus omit the index n, so that

It can be shown that the steady-state probability distribution of the queue size obtained

epochs is the same at any arbitrary time epoch (see [Gro85]) Thus, in particular, also givesthe probability distribution we are interested in, that is at customer arrival epochs

Let us define now the probability generating function (PGF) of the queue size Q

and of the customer arrivals A

After some algebraic manipulation, including application of L’Hôpital’s rule, and recalling that

Khinchin (P–K) transform equation

(A.3)

after some algebraic computations results in the well-known Pollaczek–Khinchin (P–K) value formula

=

app_que Page 393 Monday, November 10, 1997 8:55 pm

394 Synchronous Queues

However, such a P–K mean value formula can also be obtained through a simpler mean-valueargument that only uses Equation A.1 (see, e.g., [Kle75] and [Gro85]) The mean value of thequeueing delay is now obtained via Little’s formula

The average number of customers waiting in the queue for the server availability isobtained from considering that the average number of customers in the server is ρ andthe corresponding waiting time is given by Little’s formula, that is

(A.4)

(A.5)Equations A.4–A.5 are independent of the specific service order of the customers in thequeue A simple recurrence formula giving all the moments of waiting time that applies only

to a FIFO queue has been found by Takacs [Tak62]

(A.6)

A.1.1.2 The asynchronous M/D/1 queue

which the service time is deterministic and equal to θ We assume θ as the basic time unit, sothat the service time is 1 unit of time and the server utilization factor is where p

denotes the mean arrival rate of the Poisson distribution In this case the PGF of the number

of arrivals is

so that the queue size PGF (Equation A.3) becomes

Expansion of this equation into a Maclaurin series [Gro85] gives the queue size probabilitydistribution

E[ ]δ E Q -[ ]λ E[ ]θ λE θ

2[ ]

2 1( –ρ) -+

-=

(A.7)

where the second factor in is ignored for

The above Equations A.4–A.6 expressing the average queue size and delay for the provide the analogous measures for the queue by simply putting

moment given by Equation A.5:

(A.8)

as well as all the other moments for a FIFO service provided by Equation A.6:

(A.9)

A.1.1.3 The synchronous M/D/1 queue

Let us study a synchronous queue by showing its similarities with the asynchronous

queue In the synchronous queue the time axis is slotted, each slot lastingthe (deterministic) service time of a customer, so that all arrivals and departures take place atslot boundaries This corresponds to moving all the arrivals that would takeplace in the asynchronous queue during a time period lasting one slot at the end ofthe slot itself So the number of arrivals taking place at each slot boundary has a Poisson distri-

bution In particular, at the slot boundary the queue first removes a customer (if any) from the queue (its service time is complete) and then stores the new customer requests Since each cus-

tomer spends in the queue a minimum time of 1 slot (the service time), this queue will be

referred to as a synchronous queue with internal server, or S-IS queue Thequeueing process of customers observed at discrete-time epochs (the slot boundaries) is a dis-crete-time Markov chain described by the same Equation A.1 in which and representnow the customers in the queue and the new customers entering the queue, respectively, at

the beginning of slot n Thus the probability distribution found for the

the moments of the waiting time are thus expressed by Equation A.9 Therefore the first twomoments for a FIFO service are given by

=

In this queue the carried load ρ, which equals the mean arrival rate p, since the queue

capacity is infinite, is immediately related to the probability of an empty queue ,since in any other state the server is busy So,

which is consistent with Equation A.7

We examine now a different synchronous queue in which at the slot boundary

the queue first stores the new customer requests and then removes a customer request (if any) in

order for that request to be served In this case the customer is supposed to start its service time

as soon as it is removed from the queue In other words the queue only acts as a waiting linefor customers, the server being located “outside the queue” Now a customer can be removedfrom the queue even if it has just entered the queue, so that the minimum time spent in the

queue is 0 slot For this reason this queue will be referred to as a synchronous queue with external server, or an S-ES queue The queue then evolves according to the fol-lowing equation:

(A.13)

Note that in this case the queue size equals the waiting list size, since a customer leaves thequeue in order to start its service The S-ES queue can be studied analogously to the

is obtained solving again the equation in which

6 1( –p)2 -

E[ ]η2 p 2( +p)

3 1( –p)2(2–p) -

or equivalently

together with the boundary condition The PGF of the queue size is thus obtained

Expansion of this equation into a Maclaurin series [Kar87] gives the probability distribution ofthe number of customers in the queue waiting for the server availability:

(A.14)

where the second factor in is ignored for

Also in this queue the carried load ρ can be expressed as a function of the probability of anempty system However now, unlike in the S-IS queue, the server is idle in a slot ifthe queue holds no customers and a new customer does not arrive Thus

which is consistent with Equation A.14

All the performance measures of the S-ES queue can be so obtained using the

probability distribution q or the correspondent PGF In particular, by differentiating with respect to z and taking the limit as , we obtain the average number of cus-tomers in the queue, which immediately gives the average time spent in the queue , thatis

It is very interesting but not surprising that the average queueing time in the S-ES queue equals the average waiting time in the S-IS queue In fact, the onlydifference between the two queues is that in the former system a customer receives servicewhile sitting in the queue, whereas in the latter system the server is entered by a customer justafter leaving the queue

398 Synchronous Queues

A.1.2 The Geom(N)/D/1 queue

queue with the only difference lying in the probability distribution of customer arrivals at each

slot Now the service requests are offered by a set of N mutually independent customers each

request arrivals A to the queue in a generic slot is binomial:

(A.15)

with mean value p and PGF

queue (defined analogously to a S-IS queue) is characterized by the same evolutionequation (Equation A.1) as the S-IS queue, provided that the different customerarrival distribution is properly taken into account Then the PGF of the queue occupancy is

Following the usual method, the average queue occupancy is obtained:

average queue occupancy of an S-IS queue as This result is consistent withthe fact that the binomial distribution of the customer arrivals (see Equation A.15) approaches

a Poisson distribution with the same rate p as

N

-E W[ S–IS M D 1⁄ ⁄ ]+

A.1.3 The Geom/G/1 queue

A (synchronous) queue is a discrete-time system in which customer arrivals anddepartures take place at discrete-time epochs evenly spaced, the slot boundaries The arrival

process is Bernoulli with mean p , which represents the probability that a tomer requests service at a generic slot boundary The service time θ is a discrete randomvariable with general probability distribution assuming values that are integermultiples of the basic time unit, the slot interval The analysis of this queue,first developed in [Mei58], is again based on the observation of the number of customers left

(Equation A.2) apply also in this case considering that now the distribution of customerarrivals in a number of slots equal to the service time has a different expression Such a distri-bution is obtained computing the number of arrivals when the service time lasts m slots

and applying the theorem of total probability while considering that the number of arrivals in

m slots is not larger than m:

Apparently the PGF of the number of customer in the system is again given byEquation A.3 in which now

By applying twice L’Hôpital’s rule and considering that

we finally obtain

The average queueing time and waiting time are obtained applying Little’s formula:

A.1.4 The Geom/G/1/B queue

which the queue capacity, B, is now finite rather being infinite This queue is studied