IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 6, JUNE 1999
185
Dynamically Reducing Retransmission Control for Slow-Frequency-Hopped Communication Systems Katsumi Sakakibara, Member, IEEE, Kazushi Motonaga, and Yoshiharu Yuba, Member, IEEE
Abstract— A dynamically reducing retransmission control scheme is proposed for slow-frequency-hopped communication systems. In the proposed scheme a transmitter defers transmission of a new packet until all the other transmitters receive positive acknowledgments, so that the number of active transmitters are dynamically reduced. The performance of the proposed scheme is compared with the conventional scheme, in which a transmitter is permitted to transmit a new packet in any slot, in terms of the normalized throughput and the 98% packet transmission delay. The numerical results show that the proposed scheme outperforms the conventional scheme, although for some values of the number of transmitters the conventional scheme has higher normalized throughput.
I. INTRODUCTION
T
HE USE OF proper error-correcting codes enables resolution of packet collisions in frequency-hopped communication systems [1], [2]. The resolvability of packet collisions depends on the error-correcting capability of the codes employed and on the number of packets involved in a collision, since the packet error probability is certainly an increasing function the number of packets in a collision. Assuming that the number of packets in conflict is constant, Kim and Stark [2] determined the optimal coding rate for slow-frequency-hopped communications. With this assumption a transmitter is always allowed to transmit a new packet in any packet transmission slot. In this letter, we propose a dynamically reducing retransmission control scheme for slow-frequency-hopped communication systems. The performance of the proposed scheme is compared with the conventional scheme [2], in which a transmitter is permitted to transmit a new packet in any slot, in terms of the normalized throughput and the 98% packet transmission delay. In the proposed scheme a transmitter defers a new packet transmission until all the other transmitters receive positive acknowledgments. Thus, the number of packets in collision can be dynamically reduced in a certain period, which brings improvements of the packet error probability.
Manuscript received March 9, 1998. The associate editor coordinating the review of this letter and approving it for publication was Prof. B. R. Vojcic. This work was supported in part by the Ministry of Education, Science, Sport, and Culture in Japan under Grant-in-Aid for Encouragement of Young Scientists. The authors are with the Department of Communication Engineering, Okayama Prefectural University, Soja, 719-1197 Japan (e-mail:
[email protected]). Publisher Item Identifier S 1089-7798(99)05509-X.
II. SYSTEM MODEL AND PACKET ERROR PROBABILITY Consider a slow-frequency-hopped communication system pairs of transmitters and receivers, where consisting of each pair shares the identical hopping pattern. The available bandwidth is divided into frequency bins, each of which can convey an -ary signal. Every transmitter and receiver pair changes frequency bins in a symbol-by-symbol manner (slow-frequency-hopped). The time axis is divided into slots of constant length, which suffice for a packet transmission of symbols. A transmitted packet is encoded with an Reed–Solomon code over GF( ), where . Symbol errors are assumed to occur only when two or more symbol transmissions occupy the identical frequency bin, referred to as a hit. We also assume that all the transmitters can obtain a positive or negative acknowledgment (ACK or NAK) through error-free feedback channels immediately at the end of each slot. Then, the conditional probability of a symbol hit, given that transmitters are sending signals simultaneously, can be evaluated as [2] (1) where for synchronous hopping (2) for asynchronous hopping is the probability that two transmitters collide in the same frequency bin. Hence, if no side information is available at the receivers, then the packet error probability for packet collision of order is evaluated as (3) or fewer symbol errors can be corrected since with an appropriate bounded-distance decoder [3]. Note that is an increasing function the packet error probability with respect to . III. PROTOCOL DESCRIPTION A transmitter that is permitted to transmit a packet is referred to as active. In the proposed scheme, a transmitter that receives an ACK defers a new packet transmission until all the other transmitters receive ACK’s as shown in Fig. 1. In Fig. 1, Transmitter 1
1089–7798/99$10.00 1999 IEEE
186
IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 6, JUNE 1999
Fig. 1. Illustrative example of proposed scheme for
N = 8.
Fig. 3. Trellis diagram of the number of transmitters for proposed scheme.
The probability that the system is in State at the beginning of the th slot is given by (5) Fig. 2. Illustrative example of conventional scheme for
N = 8.
where
is not active in Slots 2–4, although it receives an ACK for Packet 1 in Slot 1. Packet transmissions for new packets are synchronized among transmitters. Time slots between new packet transmissions are referred to as a cycle. Two cycles for Packets 1 and 2 are shown in Fig. 1. The number of active transmitters is decreasing in a cycle. Thus, the packet error probability can be improved slot by slot. Alternatively, in the conventional scheme [2], all the transmitters are active and send their packets in every slot. Thus, packets conflict in every slot as shown in Fig. 2. IV. PERFORMANCE ANALYSIS We consider the normalized throughput [2], and the 98% packet transmission delay as performance measures. The noris defined as the average number malized throughput of transmitters which succeed in packet transmission in a slot normalized by the number of frequency bins and the . The 98% packet coding rate of Reed–Solomon codes transmission delay is the expected number of slots that are, with probability of 0.98 or more, necessary for a transmitter to receive an ACK and to be able to transmit the next packet.
if otherwise.
(6)
is an increasing function Apparently, the probability with respect to . In the trellis diagram given in Fig. 3 State 0 implies that no transmitters have packets to be retransmitted and that all the transmitters can be active in the next slot. Then, we can obtain the probability that a cycle consists of slots as
(7) . for Consequently, the normalized throughput for the controlled scheme is given by (8) While, using (7), we can obtain the 98% packet transmission delay as the smallest integer which meets (9) B. Conventional Scheme
A. Proposed Scheme In the proposed scheme, the number of active transmitters are generally decreasing in a cycle. Supposing the number of active transmitters is the state number, we can construct a trellis diagram, as shown in Fig. 3. The state transition from State to State occurs when transmitters fail to (re)transmit their packets, given that packets collide in a slot. Hence, the , can state transition probability from State to State , be evaluated as
The normalized throughput for the uncontrolled scheme is derived in [2] as (10) Apparently the number of slots necessary for a transmitter to receive an ACK is geometrically distributed with mean . Hence, the 98% packet transmission delay is evaluated as the minimum integer that satisfy the inequality
(4) for
and
. It is clear that
.
(11)
SAKAKIBARA et al.: SLOW-FREQUENCY-HOPPED COMMUNICATION SYSTEMS
Fig. 4. Normalized throughput for
q
= 100 and
n
= 32.
187
Fig. 6. 98% packet transmission delay for
q
= 100 and
n
= 32.
less than , no apparent difference can be observed between , the conventional the two schemes. For greater than , scheme excels the proposed scheme. For the proposed scheme outperforms the other. and for . Fig. 5 illustrates behavior of gradually decreases, while increases rapidly In Fig. 5 . From an asymptotic point of view and approaches to the ranges where the conventional scheme is superior to the proposed scheme seem to vanish as goes large. The 98% packet transmission delay for the same parameters as Fig. 4 is illustrated in Fig. 6. The proposed scheme requires no more slots than the conventional scheme for most values of . Hence, the proposed scheme is favorable for packet transmissions that is sensitive to transmission delay such as voice packets. VI. CONCLUSION Fig. 5. Behavior of
N0
and
N
P for
k=n
= 0:25.
It can be easily solved and we obtain (12) where
is the minimum integer not less than . V. NUMERICAL RESULTS
AND
A dynamically reducing retransmission control scheme has been proposed for slow-frequency-hopped communication systems. We have compare the performance of the proposed scheme with that of the conventional scheme in terms of the normalized throughput and the 98% packet transmission delay. The numerical results show that the proposed scheme outperforms the conventional scheme for large numbers of transmitters.
DISCUSSIONS
We present numerical results for , , and . We assume synchronous hopping and no side information at the receivers, so that the packet error probability . is given by (3) with The normalized throughput is shown in Fig. 4 with parameter . The solid and dashed lines present the performance of the proposed and conventional schemes, respectively. From , and . For Fig. 4 we can obtain two values of ,
REFERENCES [1] J. E. Wieselthier and A. Ephremides, “Discrimination against partially overlapping interference—Its effects on throughput in frequency-hopped multiple access channels,” IEEE Trans. Commun., vol. COM-34, pp. 136–142, Feb. 1986. [2] S. W. Kim and W. Stark, “Optimum rate Reed–Solomon codes for frequency-hopped spread-spectrum multiple-access communication systems,” IEEE Trans. Commun., vol. 37, pp. 138–144, Feb. 1989. [3] S. B. Wicker, Error Control Systems for Digital Communication and Storage. Englewood Cliffs, NJ: Prentice-Hall, 1995.
