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Chia-Hung Wei, Ping-Chen Lin, and Ray-Guang Cheng, Senior Member, IEEE. Abstract—In [1], the throughput and access delay performance of the RACH in ...
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 1, JANUARY 2013

Comment on ‘An Efficient Random Access Scheme for OFDMA Systems with Implicit Message Transmission’ Chia-Hung Wei, Ping-Chen Lin, and Ray-Guang Cheng, Senior Member, IEEE

Abstract—In [1], the throughput and access delay performance of the RACH in an orthogonal frequency division multiple access (OFDMA) system is analyzed. However, the analysis of access delay is incorrect. This note corrects the original equations and results show that the new equations correct the non-negligible error of the average access delay under the uniform backoff policy. Index Terms—Access delay, binary exponential backoff, uniform backoff.

I. A NALYSIS N [1], the authors analyzed the throughput and access delay performance of random access channel (RACH) in an orthogonal frequency division multiple access (OFDMA) system under binary exponential and uniform backoff policies. They considered the case that the random access durations (RADs) are staggered in the data slots, and each RAD consists of a number of random access slots (RASs). Following the definition given in [1], denotes the duration of each RAS as TS ; the number of the RASs in every RAD as M ; and the interval between two adjacent RADs as TI , as illustrated in the upper part of Fig. 1. As in [1], let Wi,B and Wi,U be the backoff delay in the ith retransmission for the binary exponential and uniform backoff policies, respectively; ps be the successfully transmission probability; R be the number of retransmissions during the random access and r be the retransmission times; DB and DU be the random variable denoting the access delay under the binary exponential and the uniform backoff policies, respectively; and DB and DU be the average value of DB and DU , respectively. In [1], DB and DU were derived from the conditional expected access delay multiplied by the probability for the number of retransmissions. However, the conditional expected access delay of E[DB |R = r > 0] and E[DU |R = r > 0] given in [1, Eq. (8)] and [1, Eq. (11)] and the average access delay of DB and DU given in [1, Eq. (9)] and [1, Eq. (12)], respectively, have some errors.

I

A. Delay Analysis under Binary Exponential Backoff Policy In binary exponential backoff policy, ωi and Ui are uniformly distributed in [1, 2i−1 ] and [1, M ], respectively. The Manuscript received June 18, 2012; revised October 2, 2012; accepted October 16, 2012. The associate editor coordinating the review of this paper and approving it for publication was G. Bianchi. This work was supported by the National Science Council, Taiwan, under Contract NSC101-2219-E011-005, NSC 101-3113-P-011-003, and NSC 1012219-E-009-026. C. H. Wei, P. C. Lin, and R. G. Cheng (corresponding author) are with the Department of Electronic and Computer Engineering, National Taiwan University of Science and Technology, Taiwan (e-mail: [email protected], [email protected], [email protected]). Digital Object Identifier 10.1109/TWC.2012.112012.120872

MTS

TI

RAD

RAD

RAD

BW1

RAD

RAD

RAD

BW2

Access delay (Case 1)

RAD

RAD

BW3

Access delay (Case 2) Successful transmission delay

Case 1 ( ω2 = 1)

U2TS

ω1 = 1 Backoff delay for the first retransmission

Case 2 (ω2 = 2) Backoff delay for the second retransmission

W1,B = ω1(MTS+TI)

W2,B = ω2(MTS+TI)

Fig. 1. Binary exponential backoff policy in OFDMA system for r = 2 [1, Fig. 3(a)].

MTS

TI

RAD

RAD

RAD

BW1

BW2

RAD

BW3

RAD

RAD

...

Access delay Successful transmission delay ω1 = 1

ω2 = 1

Backoff delay Backoff delay for the first for the second retransmission retransmission W1,B = ω1(MTS+TI)

U2TS

W2,B = ω2(MTS+TI)

Fig. 2. Uniform backoff policy in OFDMA system for r = 2 [1, Fig. 3(b)].

access delay under binary exponential backoff policy can be derived by considering the case of two retransmissions (r = 2) as illustrated in Fig. 1. In this case, ω1 = 1 and ω2 is uniformly distributed in [1, 2]. In Fig. 1, W1,B = ω1 (M TS + TI ) and W2,B = ω2 (M TS + TI ) represent the backoff delay for the first retransmission and the second retransmission, respectively, and U2 TS is the successful transmission delay after two retransmissions. Hence, the general form of the conditional expected access delay under binary exponential backoff policy should be corrected to be E[DB |R = r > 0] =

c 2013 IEEE 1536-1276/13$31.00 

=

r  i=1 r

E[ωi ](TI + M TS ) + E[Ur ]TS

(2 + r − 1) (TI + M TS ) + E[Ur ]TS . 2 (1)

WEI et al.: COMMENT ON ‘AN EFFICIENT RANDOM ACCESS SCHEME FOR OFDMA SYSTEMS WITH IMPLICIT MESSAGE TRANSMISSION’

B. Delay Analysis under Uniform Backoff Policy

250

In uniform backoff policy, ωi is a constant value (i.e., ωi = 1 in [1]) and Ui is uniformly distributed in [1, M ]. The access delay under uniform backoff policy can be obtained by considering the case of two retransmissions (r = 2) as illustrated in Fig. 2. In this case, ω1 = ω2 = 1. In Fig. 2, W1,B = 1 × (M TS + TI ) and W2,B = 1 × (M TS + TI ) represent the backoff delay for the first retransmission and the second retransmission, respectively, and U2 TS is the successful transmission delay after two retransmissions. Thus, the general form of the conditional expected access delay under uniform backoff policy is corrected to be

Sim (T = 0) I

Sim (TI = 20) Sim (TI = 40)

Access delay expectation (D)

200

New−Ana (TI = 0) New−Ana (T = 20) I

New−Ana (TI = 40)

150

Old−Ana (T = 0) I

Old−Ana (TI = 20) Old−Ana (T = 40) I

100

50

0 0.5

0.6 0.7 0.8 0.9 Successful transmission probability (ps)

1

E[DU |R = r > 0] =

r 

(TI + M TS ) + E[Ur ]TS

i=1

= r(TI + M TS ) + E[Ur ]TS . Fig. 3.

Average access delay under binary exponential backoff policy.

90 Sim (TI = 0)

Access delay expectation (D)

80

r=0

Sim (TI = 40)

=

New−Ana (TI = 0) New−Ana (TI = 20)

60

New−Ana (TI = 40) 50 40

II. N UMERICAL R ESULTS For the purpose of comparison, we plotted the average access delay of the binary exponential backoff policy and the uniform backoff policy in Figs. 3 and 4, respectively, for the same system parameter values used in [1]. Computer simulations were conducted to verify the accuracy of the analysis derived in Eqs. (2) and (4). The simulation results were coincided with the new analytical results. Fig. 3 shows that the analytical values obtained from Eq. (2) are slightly smaller than that obtained from [1, Eq. (9)]. The difference is quite marginal because practically E[Ui ]TS will be small and can be neglected. Fig. 4 shows that the analytical value obtained from Eq. (4) is much smaller than that obtained from [1, Eq. (12)] when TI is relatively larger than TS . The error in the original equation is non-negligible and may up to 100% for ps = 0.5.

20 10

0.6 0.7 0.8 0.9 Successful transmission probability (ps)

1

Average access delay under uniform backoff policy.

Following the assumption of P {R = r} = ps (1 − ps )r made in [1, Eq. (6)], DB is corrected to be =

E[DB ] =

∞  r=0

=

(4)

Old−Ana (TI = 20)

30

DB

1 − ps (1 + M ) TS . (TI + M TS ) + ps 2

Old−Ana (TI = 0) Old−Ana (TI = 40)

0 0.5

(3)

Following the same assumption in [1, Eq. (6)], DU is corrected to be ∞  DU = E[DU ] = E[DU |R = r]P {R = r}

Sim (TI = 20)

70

Fig. 4.

415

E[DB |R = r]P {R = r}

(TI + M TS ) 4ps − 3p2s − 1 (1 + M ) + TS . (2) 2 ps (2ps − 1) 2

R EFERENCES [1] P. Zhou, H. Hu, H. Wang, and H. H. Chen, “An efficient random access scheme for OFDMA systems with implicit message transmission,” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp. 2790–2797, July 2008.