1. A Note on Theoretical Limit of Type-I Hybrid Selective-repeat ARQ with Finite Receiver Buffer
Yasunari MAEDA (Kitami Institute of Technology)
Hideki YOSHIDA (Kitami Institute of Technology)
Toshiyasu MATSUSHIMA (Waseda University)

2. Topics 1. Introduction 2. Preparations 3. Modeling
4. Proposed algorithm
5. Experiments
6. Conclusion

3. 1. Introduction overview of type-I hybrid selective-repeat ARQ
codeword
received block
binary symmetric
channel
transmitter
receiver
feedback channel
noiseless
acknowledgement
symbol
ACK (a positive acknowledgement symbol)
no bit error or correctable bit errors in the received block
NAK (a negative acknowledgement symbol)
uncorrectable bit errors in the received block

4. × 1. Introduction △ × undetectable errors △ buffer over flow
example of operation in type-I hybrid selective-repeat ARQ 7 6 2 5 4 3 1 × △
× undetectable errors　 △ buffer over flow ○
round-trip delay D=3
transmitter
receiver
receiver buffer
size of
time 10 9 8
ACK
NAK
C=4
○ delivered to the user

5. 1. Introduction overview of Markov decision processes reward
transition probability
action
set of states
state
state
set of actions
a probability of an event that a state transition from state to state is occurred when action is selected at state
a true parameter of transition probability (known)
a reward of a state transition from state to state under the condition that action is selected at state
target : to maximize total rewards

6. 1. Introduction the theoretical limit of throughput of type-I hybrid
Selective-repeat ARQ
the theoretical limit of throughput with infinite receiver buffer
many previous research
the theoretical limit of throughput with finite receiver buffer
our research
In previous research many type-I hybrid selective-repeat ARQ methods with finite receiver buffer are proposed.
But the throughput of the proposed methods can not be compared with the limit with finite receiver buffer.
We study the theoretical limit of throughput of type-I hybrid selective-repeat ARQ with finite receiver buffer.

7. 2. Preparations(1/4) a codeword which a transmitter wants to send
a set of codewords
a positive acknowledgement symbol
a negative acknowledgement symbol
an acknowledgement symbol received by the transmitter at time
the minimum number of a codeword whose ACK symbol is not received yet at time
the number of codeword which is transmitted at time
the size of finite receiver buffer
a round-trip delay
(The transmitter receives the acknowledgement symbol for
at time )

8. 2. Preparations(2/4) (1) (2) (3) （actions in MDP）
a state at time in Markov decision processes(MDP)
(1)
where,
represents a situation of the receiver buffer.
(2)
represents codeword already sent by the transmitter.
(3)
（actions in MDP）

9. 　2. Preparations(3/4)
a bit error rate of a BSC(binary symmetric channel)
the probability of an event that a receiver sends back ACK symbol
a parameter which dominates
the true parameter which is known
(4)
(5)
assumptions
(n,k) linear code is used, feedback channel is noiseless,
where,
the maximum number of correctable bit errors
the number of bit errors in the received block
the maximum number of detectable bit errors

10. 2. Preparations(4/4) (6) operations from time to time
(after deciding action at time )
Transmitter receives an acknowledgement symbol at time
( is for codeword sent at time )
(2) State at time and are computed by using state at time ,
action at time and acknowledgement symbol at time
(3) Reward at time is computed by using state at time , action
at time and state at time
ACK symbol is received and
some codewords are sent to
the user ;
else .
(6)
(4) Action at time is decided.

11. 3. Modeling(1/2) modeling to solve periods MDP problem
utility function
(7)
where,
decision function which returns the number of codeword to send by using the state and time
reward after time
eqs.(7) is equal to the number of codewords which are delivered to the user.

12. 3. Modeling(2/2) expected utility function (8)
where, is first state and known.
expected value for the number of codewords which are delivered to the user
In order to compute the theoretical limit with finite receiver buffer, we have to maximize the expected utility.

13. 　4. Proposed algorithm
compute the maximum of eqs.(8) by using dynamic programming method
(9)
where, is the maximum of expected utility after time
(Expected utility is equal to the expected value for the number of codewords which are delivered to the user.)
the theoretical limit of throughput
with finite receiver buffer

14. 5. Experiments(1/2) conditions a round-trip delay D=4
size of receiver buffer C=4, C=8
number of sending codewrods T=1000
bit error rate 0.001, 0.01
using (255,115)BCH code，3 conditions
, ,
:the maximum number of correctable bit errors
:the maximum number of detectable bit errors
ISR : theoretical limit with infinite receiver buffer（ ）
PRO : proposed theoretical limit with finite receiver buffer
WEL : an averaged throughput of 1000 times simulations by
Weldon’s method
(Weldon’s method is one of typical selective-repeat ARQ methods. And we apply Weldon’s method to type-I hybrid for the experiments.)

15. 　5. Experiments(2/2)
The differences between ISR and WEL are big, but the differences between PRO and WEL are small.
case of
bit error rate
0.001
0.01
ISR
PRO
WEL
The performance of WEL is close to the theoretical limit with finite receiver buffer.
Our proposed theoretical limit with finite receiver buffer is useful to investigate the performance of previous methods.

16. 6. Conclusion We applied Markov decision processes to type-I hybrid
selective-repeat ARQ with finite receiver buffer.
We proposed the algorithm which computes the theoretical
limit of throughput of type-I hybrid selective-repeat ARQ
with finite receiver buffer.
The theoretical limit of throughput computed by our
proposed algorithm is useful to investigate the performance
of previous type-I hybrid selective-repeat ARQ methods
with finite receiver buffer.
As further works, we want to study the relationship of
throughput and decoding error probability.