THROUGHPUT PERFORMANCE OF SOME CONTINUOUS ARQ SCHEMES WITH MEMORY

much when p e r = 0.001, but increase remarkably when p e r = 0.01, especially in the region larger than U = 1.0. The volume of buffer memory to be prepared for resequencing becomes greater compared with the case of no transmission errors.

C o n c l u s i o n s : The probability of resequencing delay, the resequencing delay time, and the total delay time in multilink systems have been quantitatively calculated by simulation. It has been shown that they may increase remarkably when transmission errors occur over the link.

Y. YOSHIDA 14th September 1990

Department oJ Electrical Engineering Hosei University

Kajino-cho, Koganei-shi, T o k y o 184, Japan References

CCITT: 'Data communication networks: Services and lacilities, Interlaces', Blue Book, Vol. VIII-Fascicle VIII.2 (Geneva, 1989),

CCITT: 'Data communication networks: transmission and signal- ling and switching, network aspects, maintenance and adminis- trative arrangements', Blue Book, Vol. VIII-Fascicle VI11.3, Geneva, 1989, pp. 175- 185

NISHIZONO, I., and VOSHIDA, v.: 'An analysis on multilink system', Trans. I E C E Japan, 1 9 8 4 , 5 6 8 , pp. 47-53

pp. 182-191

THROUGHPUT PERFORMANCE OF SOME CONTINUOUS ARQ SCHEMES WITH MEMORY

Indexing term.. Telecommunications, Data transmission, A R Q techniques, Throughput efficiency

0 c r I 1. Notice that r = 1 corresponds to the memoryless case studied by Bruneel and Moeneclaey.' Small values of r can be achieved by using the soft error

Two different demodulation strategies proposed in Refer- ence 1, namely the limited information strategy (LIS) and the full information strategy (FIS), are considered. In the case of LIS, the receiver makes a decision on a given data block by combining only the current set of copies of the data block regardless of all the information gained from the previous sets. The FIS combines all copies received thus far to make a deci- sion on a given data block even though these copies were received from different sets of copies. Clearly, FIS requires more memory than LIS does. The throughput efficiency of both strategies will be derived and optimised, and compared with that of the corresponding schemes without memory studied by Bruneel and Moeneclaey in Reference 2.

T h r o u g h p u t e v a l u a t i o n a n d o p t i m i s a t i o n : Let B(no, n , , . . .) denote the mean number of transmissions sent per correct data block delivered. The throughput efficiency of an ARQ scheme is = l/B(n,, n,, ...). Let n: denote the optimum value of the parameter n, ( i 2 0). That is

B ( n f , n:, .. .) 2 B(n,, n,, ...) Yno, ni, . . .) (1) For simplicity, we assume that the round-trip delay is s, the number of data blocks that can be transmitted during the interval between the end of transmission of a copy and the receipt of its response is constant, and all the data blocks are of fixed length. Expressions for the quantity B(no, ni,

. .

LIS:

B(no, n i , ...) = a(no)

+

+

..)I

I n f l < s + l

A class 01 continuous ARQ schemes with memory, where multiple copies 01 each data block are sent contiguously instead of one single copy, and several copies of the same data block are combined to make a decision at the receiver, is considered. The throughput eficiency analysis is carried out based on the model proposed by Bruneel. Results show that the performance 01 the investigated ARQ schemes can be considerably improved if the receiver has some memory.

I n t r o d u c t i o n : Bruneel and Moeneclaey' analysed the through- put efficiency of a class of continuous ARQ schemes in which each data block is transmitted no or less copies and, if neces- sary, retransmitted n, or less copies in the ith retransmission to the receiver. This class of ARQ schemes preserves the ord- ering of data blocks just as in the classic go-back-N ARQ scheme. It is possible that an ACK for a copy of a given data block arrives at the transmitter before all the maximum allowed number of copies are sent. When this occurs, the transmitter will start to transmit the next data block instead of continuing to transmit the other copies of the data block. This is the reason the phase 'or less' is used in the above description. An error detection procedure is performed at the receiver on each received copy independently.'

In this letter, we allow the receiver to have some memory, i.e., the receiver can combine several copies to make a more reliable decision. The analytical model proposed by Bruneel' is used for performance evaluation. This analytical model is as follows: Whenever a data block is transmitted for the first time after no data blocks ahead of it will even be retransmitted, the erroneous decision probability is given by p . When two or more copies of the same data block are sent to the receiver, the information contained in all (or parts) of these copies can be collected to obtain a more reliable decision. In the model, this process results in a constant reduction factor, say r , of the erroneous decision probability for each additional received copy. In other words, the probability that the j t h copy of a demodulated data block is erroneous given that its preceding j ~ 1 copies are all erroneous is equal to p F i ,

One can easily prove that all the n:s are equal to a common optimum value n*. As a result, the optimum B(n,, n , , . . . ) is given by

F I S :

&no, n , , ... ) = ao(no)

+

+

)I

and N , = n, ( i 2 0). Notice that all the n:s are no longer equal to a common optimum value. In order to simplify the implementation, it is of interest to consider the special case when no = ni = . . . = n. For this special case, eqn. 5 reduces to

drf (s

+

dnrJ"""

I

+

r

,Io

ELECTRONICS LETTERS 8th November 1990 Vol. 26 No 23

where n - s - I B i n , n, . . .) = s

+

1

IS

+

.)I

For convenience, we refer to the special case as the fixed FIS (FFIS). In the cases of both FIS and FFIS, it is difficult to obtain a further explicit expression of the optimum B(n,, n,, . . .). but we can still resort to computer searching.

Results and d i s c u s s i o n : The results presented in Reference 2 can be found by substituting r = 1 into either eqn. 4 or 8. I n the case of FIS (including FFIS), a practical computation of results requires an approximation of the infinite series in eqns.

5 and 8 by a finite sum.

07 0 6 L 3 a & 0 3 - 0 L, 3 0 2 -

A R Q without memory

',

0 1 l - I I

0 0 0 2 O L 0 6 0 8 1 0

p (block error probablllty)

Fig. 1 Throughput efficiency against p s = 10; r = 0.8

Figs. 1 and 2 show the throughput efficiency against p for

s = IO, r = 0.8, and s = 50, r = 0.6, respectively. Here the throughput efficiency is achieved by using all the optimum values of n,, and the logarithmic scale is used in the vertical axis of both Figures. It can be seen that, under a wide range of

high error rate conditions, considerable improvements in

C O 0 2 0 4 0 6 0 8 ' , G

p (block e r r o r p r o r a b r h t y )

Fig. 2 Throughput efficiency against p s = 50; r = 0.6

throughput performance can be achieved in both cases if the receiver has some memory. One can also observe that as expected, FIS performs better than FFIS, and LIS is the most inefficient technique. The performance difference among FIS, FFIS and LIS can be neglected especially in environments with a large round-trip delay. The curve of the optimum number of copies as a function of p for s = 20 and r = 0.8 is plotted in Fig. 3. This Figure shows that the optimum copy

201

J

I

A

0 2 0 4 0 6 0 8 1 0

Fig. 3 Optimum number ofcopies againSt p s = 20; r = 0-8

number can be reduced significantly for high values of p by adding memory in the A R Q schemes. We conclude that LIS

policy should be a good choice in practice.

T-H. LEE J.-K. HU'

Department of Communications Engineering 'Institute of Electronics

National Chiao Tung University Hsinchu, Taiwan 30050 Republic ofChina References

28th September 1990

BRUNEEL, H . : 'Throughput comparison for stop-and-wait ARQ

schemes with memory', Electron. Lett., 1988, 24, pp. 531-533

BRUNEEL, H., and MOENECLAEY, M . : 'On the throughput per-

formance of some continuous ARQ strategies with repeated trans- mission', / € € € Trans., 1986, COM-34, pp. 244249

FANTAKI. R.: 'Generalised stop-and-wait ARQ scheme with soft

error detection'. Electron. Lett., 1986, 22, pp. 882-883

LAU, c., and LEUNG, c.: 'Performance analysis of a memory A R Q

scheme with soft decision detectors'. IEEE Trans., 1986, COM-34,

pp 827-832

SPLIT-RADIX FHT ALGORITHM FOR

REAL-SY M M ETRlC DATA Indexing term Transforms

A fast algorithm for computing the discrete Hartley trans- form of a real-symmetric data sequence is introduced. The number of computations required is significantly less than that required by the usual split-radix fast Hartley transform.

I n t r o d u c t i o n : A number of different algorithms have been pro- posed for computing the discrete Hartley transform (DHT) which are called fast Hartley transform ( F H T )