An exact closed-form formula of collision probability in diverse multiple access communication systems with frame slotted aloha protocol

Accepted Manuscript

An Exact Closed-Form Formula of Collision Probability in Diverse Multiple Access Communication Systems with Frame Slotted Aloha Protocol Kun-Shu Huang, Chi-Kuang Hwang, Bore-Kuen Lee, In-Hang Chung PII: DOI: Reference:

S0016-0032(17)30263-6 10.1016/j.jfranklin.2017.05.028 FI 3002

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

14 August 2016 16 April 2017 20 May 2017

Please cite this article as: Kun-Shu Huang, Chi-Kuang Hwang, Bore-Kuen Lee, In-Hang Chung, An Exact Closed-Form Formula of Collision Probability in Diverse Multiple Access Communication Systems with Frame Slotted Aloha Protocol, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.028

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An Exact Closed-Form Formula of Collision Probability in Diverse Multiple Access Communication Systems with Frame Slotted Aloha Protocol

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Kun-Shu Huanga,∗, Chi-Kuang Hwangb , Bore-Kuen Leeb , In-Hang Chungb a Ph.D.

Program in Engineering Science, College of Engineering, Chung Hua University, Hsinchu 30012, Taiwan, ROC b Department of Electrical Engineering, Chung Hua University, Hsinchu 30012, Taiwan, ROC

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Abstract

For diverse multiple access communication systems based on frame slotted aloha (FSA) protocol, it is important to analyze collision probability for the system performance evaluation. As shown in the literature, for general settings, it is difficult to derive an exact and closed-form solution for collision probability

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without approximation. Recently, an exact solution based on generic analytical approach (GAA) [31] has been proposed, yet its numerical computation will

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become difficult when the number of slots is larger than 16. In this paper, we develop an exact closed-form formula (ECFF) for collision probability that can not only overcome the computational deficiency of GAA in the presence of a

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large number of slots, but also reduce the computation complexity of collision probability. Surprisingly, by introducing a differentiation operator to form a hybrid recursive equation and applying various algebraic properties of Laplace

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transform and Z transform, the final collision probability can be represented by a compact double summation. Accuracy of the ECFF and comparison with the

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GAA have been studied by Monte Carlo simulation. Keywords: Collision probability, Differentiation operator, Hybrid ∗ Corresponding

author. Tel.: +886 3 5186396; fax: +886 3 5186436 Email addresses: [email protected] (Kun-Shu Huang), [email protected] (Chi-Kuang Hwang), [email protected] (Bore-Kuen Lee), [email protected] (In-Hang Chung)

Preprint submitted to Journal of The Franklin Institute

May 25, 2017

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equation, Laplace transform, Z transform.

1. INTRODUCTION

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When two or more signals arrive in the same shared wireless channel simul-

taneously, the interferences among these signals can result in collision detection

in the receiver. As a result, the efficiency of the signal transmission will be 5

significantly reduced due to the unwanted collision. The framed slotted aloha (FSA) protocol has been proposed as an anti-collision scheme to improve the

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transmission efficiency in communication networks. For the FSA-protocol-based

wireless communication systems, multiple users transmit signals to a common receiver, for examples, the base stations on the ground and the transponder in a 10

satellite network [1], the users and the base station in a mobile radio network [2], the tags and the reader in the Radio Frequency Identification (RFID) network [3-17]. The number of slots adopted within a frame of the FSA protocols can

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be fixed or even varied dynamically frame by frame. Every user can randomly transmit a packet into any slot so that the transmission problem of a frame is 15

similar to the occupancy problem of distinguishable balls to be randomly placed

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into distinguishable slots [18]. There are only three possible outcomes for each slot in a frame, which are “single” with one packet, “empty” without any packet,

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and “collision” with more than one packet to be allocated in it. After EPC Global establishing the standard for RFID systems [21], many 20

researchers have studied numerous FSA-scheme-based algorithms with either

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fixed or dynamic frames for identifying object or resolving collision in RFID systems [8-16, 22-29]. The essential jointed probability P[S,E,C] for the event in

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the frame with S “single” slots, exact E “empty” slots, and C “collision” slots has been developed [9-17, 30]. An approximation form of the P[S,E,C] based on

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the Poisson arrival statistics has been derived in [10]. Several simple analytic forms of the P[S,E,C] have also been derived and adopted in [9-11]. Then, the

improper assumption in [9-11] that the outcomes of the slots within a frame are independent has been revised in [12-13]. Besides, our previous work in [31] has

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pointed out that there are still some mathematics errors in the revised versions 30

[12-13]. Estimation of the number of “collision” slots in a frame may not be a deadly

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issue for communication scenarios where users can resend the signal or message. However, the collision probability will become very crucial for traffic manage-

ment. The probability P[k] for the event that there are k “collision” slots in a 35

frame has been developed based on generic analytical approach (GAA) in the FSA schemes [31]. Therein, a three-stage approach was proposed to consider

the three stages, particular k “collision” slots, the sum of the particular k “col-

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lision” slots, and the exact k “collision” slots, respectively. Since the number of summation indices will be the same with the number of particular “collision” 40

slots at the first stage of GAA, a large number of binomial coefficients contained in the mathematical expressions of the GAA will cause difficulty to calculate probability P[k] when R and/or L becomes large, where R and L are the numbers of balls and slots, respectively. Especially, for large L such as L ≥ 16, the

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computation burden in GAA is usually over the capacity limit of normal computing facilities to result in unacceptable round-off errors. Moreover, if further

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characteristics are to be extracted from the probability P[k] , a neat closed form for P[k] will be required.

In the paper, we shall reformulate the mathematical expressions of GAA to

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develop an exact closed form formula (ECFF) for probability P[k] . By using the number of combinations, that k particular slots are “collision” for R balls and L

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slots, as the new dependent variable at the first stage in GAA, a new recursive equation will be formulated to overcome the problem of infeasible number of summation indices at the first stage. Additionally, one differentiation operator

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will be introduced in this recursive equation so that variable R has nothing to

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do with recursion and thus can be omitted. Thus, the recursive system will become a hybrid one which contains both a continuous-time variable L and a discrete-time variable k. Then Laplace transform will be adopted to solve the continuous-time subsystem, while the Z transform will be used to resolve the discrete-time recursive subsystem. Then, for the probability of the event that 3

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k particular slots are “collision”, it can be represented as a double summation, instead of many repeated summations with at most L indices in GAA. For the further analysis, we establish an alternative expression for such a probability,

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that contains only a single summation, but with an additional differential operator. The probability of the exact k “collision” slots at the third stage induces 65

another summation, so the final probability is a double summation with one binomial differential operator. A novel algorithm based on power series will be proposed in this paper, and it can simplify a double summation containing

one binomial differential operator to be a neat double summation without any

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differential operator.

The rest of the paper is organized as follows. The problem formulation is

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given in Section 2. The GAA is revisited in Section 2.1 and the proposed ECFF for the collision probability is developed in Section 2.2. In Section 3, verification and comparison of the two analytical algorithms are made through numerical

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made in Section 4.

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simulation by using the Monte Carlo method. Finally, some conclusions are

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2. PROBLEM FORMULATION

Consider the occupancy problem of R distinguishable balls to be randomly

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placed into L distinguishable slots [18]. The three-stage approach of the GAA [31] under the FSA scheme will be first revisited with a clear definition in the 80

sequel. For the simplicity of representing the idea, we take an example of three

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slots so that there are seven possible “collision” cases as listed in Table 1. At the first stage, the event that a set of k particular slots are “collision” is considered, and the probability of this event is defined as P (k, L) = w(k, L)/LR where

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w(k, L) is the number of combinations of the event at this stage. Without loss

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of generality, we can calculate P (k, L) under the assumption that the first k

slots are “collision”. Then, at the second stage, the event that at least k slots are “collision” is considered, and the probability of this event is defined as Sk

which is the sum of the probabilities of all possible equal-likely events at the first

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stage. Finally, at the third stage, the event that exact k slots are “collision” is 90

considered, and the probability of this event is denoted by P[k] . For three slots,

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all possible events at the three stages are shown in Table 2. Table 1: Possible collision cases based on three slots Case

A

Slot 1

“collision”

B

Slot 2

D

“collision”

G

“collision”

“collision” “collision”

H

“collision”

J

“collision”

“collision”

“collision”

“collision”

“collision”

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Slot 3

F “collision”

Table 2: Possible cases for three probabilities The first stage

The second stage

The third stage

Slot 1 is “collision”: {A,F,G,J}

At least one “collision” slot:

Exact one slot is “collision”:

P (1, L) = w(1, L)/LR = 4/LR

{A,B,D,F,G,H,J}, S1 = 7P (1, L)

{A,B,D} with probability P[1]

Slot 1 and 2 are “collision”: {F,J}

Exact two slots are “collision”:

{F,G,H,J}, S2 = 4P (2, L)

{F,G,H} with probability P[2]

Slot 1, 2 and 3 are “collision”: {J}

At least three “collision” slots:

Exact three slots are “collision”:

{J}, S3 = P (3, L)

{J} with probability P[3]

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P (3, L) = w(3, L)/LR = 1/LR

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At least two “collision” slots:

P (2, L) = w(2, L)/LR = 2/LR

2.1. Generic Analytical Approach (GAA)

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The generic analytical approach (GAA) is revisited in this section with a clearer derivation. According to the three stages described previously, we start 95

with the probability P (k, L), then the sum of the probability Sk follows, and

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end with the probability P[k] . Let P (1, L) denote the probability of the event that a particular slot in the frame is “collision”. This slot can be anyone of the

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L slots, but we define it as the first slot without loss of generality. Then P (1, L) can be expressed as

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P (1, L)

R   R X X R (L − 1)R−n1 R! (L − 1)R−n1 = = .(1) n1 LR (n1 )!(R − n1 )! LR n =2 n =2 1

1

Furthermore, let P (2, L) denote the probability of the event that any two particular slots in the frame are “collision”. Then, we denote such two slots as 5

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the first and the second ones, so P (2, L) can be written as R−2 X1 R − n1  (L − 2)R−n1 −n2 X  R  R−n P (2, L) = n2 n1 n =2 LR n =2 2

X

=

n1 =2

X

n2 =2

R! (L − 2)R−(n1 +n2 ) . (2) (n1 )!(n2 )!(R − n1 − n2 )! LR

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1

R−2(2−1) R−n1

Likewise, let P (3, L) denote the probability for the event that any three particular slots in the frame are “collision”. Then, P (3, L) can be written as R−2(3−1) R−2(3−2)−n1 R−(n1 +n2 )

P (3, L)

X

=

X

n1 =2

X

n2 =2

n3 =2

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R! (L − 3)R−(n1 +n2 +n3 ) . (3) (n1 )!(n2 )!(n3 )!(R − n1 − n2 − n3 )! LR

Inductively, let P (k, L) denote the probability for the event that any k par-

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ticular slots are “collision”. Then P (k, L) can be represented as

R−(n1 +n2 +...+nk−1 )

R−2(k−1) R−2(k−2)−n1 R−2(k−3)−(n1 +n2 )

X

P (k, L) =

X

n1 =2

n2 =2

X

X

...

n3 =2

nk =2

R−(n1 +n2 +...+nk )

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R! (L − k) (n1 )!(n2 )!...(nk )!(R − n1 − n2 − ... − nk )!

.

LR

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Note that the order of the k slots defined in P (k, L), 2 ≤ k ≤ L, is irrelevant. Define Sk as the sum of the probability P (k, L) for all equal-probable events

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that k slots in a frame are “collision”. That is,     L L Sk = P (k, L), with S0 = P (0, L) = 1. k 0

(5)

Note that actually Sk is the probability of the event that at least k slots are

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“collision”. Then, using the principle of inclusion and exclusion [18], we can obtain the probability for the event that there are exactly k “collision” slots in

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a frame, P[k] as P[k]



     k+1 k+2 L−k L = Sk − Sk+1 + Sk+2 + ... + (−1) SL . (6) k k k

Note that by using Eqs. (4)-(6), the probability of the event that there is no

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“collision” slot, P[0] can be obtained as P[0]

= S0 − S1 + S2 + ... + (−1)L SL = 1 − 6

L X

k=1

P[k] .

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Theoretically, following the GAA described above and employing Eqs. (4)(6), we can evaluate P[k] straightforwardly. However, it can be observed that there are a large number of binomial coefficients or factorials involved in the

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algebraic expressions. As R and/or L becomes large, especially for large L, the calculation of P[k] becomes infeasible due to capacity limit and round-off errors

of computing facilities. Moreover, it is always desirable to obtain a closedform expression for a mathematical model so that we can easily observe some properties of the model. In the next section, we will develop the ECFF of P[k] .

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2.2. An Exact Closed Form Formula (ECFF) for GAA

At first we will define the number of combinations that k particular slots are

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“collision” for R balls and L slots as w(k, L, R) = LR P (k, L). Then the total number of all possible combinations without considering “collision” is given as w(0, L, R) = LR . Consider the situation that a particular slot, which is without loss of generality the first slot is “collision” and such a situation is equivalent to the complement of the following two cases: the first case is that the particular

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slot is occupied by one ball, and the associated number of combinations is

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R(L − 1)R−1 ; the second one is that the particular slot is empty and its number of possible combinations is (L − 1)R . Thus, the number of combinations that

a particular slot is “collision”, i.e., k = 1, can be represented as the following basic two-dimensional recursive form, which is different from the GAA [31].

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w(1, L, R)

= LR − (L − 1)R − R(L − 1)R−1

= w(0, L, R) − w(0, L − 1, R) − Rw(0, L − 1, R − 1).

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Normally, Z transform is the fundamental skill to deal with the recursive

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form, but it will encounter a complex two-dimensional recursive equation for general k “collision”. To avoid such a problem of Z transform, a novel technique is proposed by introducing the differentiation operator DL = ∂/∂L, so that

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DL w(0, L − 1, R) = Rw(0, L − 1, R − 1) and thus Eq. (8) can be rewritten as w(1, L, R)

= w(0, L, R) − w(0, L − 1, R) − DL w(0, L − 1, R − 1). 7

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Since variable R has nothing to do with recursion in the last equation, we can omit R from the last equation, i.e., abbreviating w(k, L, R) to w(k, L), to obtain a more compact recursive form w(1, L) = w(0, L) − w(0, L − 1) − DL w(0, L − 1), 145

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which is a hybrid system containing both continuous-time and discrete-time operations.

Now we consider the case that two particular slots, specifying the first slot

and the second slot, are “collision”. Based on the above compact recursive form

and the idea about the complement of both of the only one ball case and empty case, the number of combinations that two particular slots are“collision” can be written as

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w(1,L)

w(2, L)

}| { z = LR − (L − 1)R − R(L − 1)R−1 w(1,L−1)

−

DL w(1,L−1)

z }| { [R(L − 1)R−1 − R(L − 2)R−1 − R(R − 1)(L − 2)R−2 ] . (10)

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−

z }| { [(L − 1)R − (L − 2)R − R(L − 2)R−1 ]

Therefore, by induction, for the case that k particular slots are “collision”, its

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general recursive form can be formulated as w(k + 1, L)

= w(k, L) − w(k, L − 1) − DL w(k, L − 1).

(11)

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We review some basic formulas of Z transform and Laplace transform for the hybrid system Eq. (11) which includes both continuous-time and discretetime operations. Thereby, we can solve the hybrid system Eq. (11) as stated in ∞ P Theorem 1. With the definition of Z transform Z{x(k)} = x(k)z −k = X(z),

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155

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the transform of the time difference function is Z{x(k + 1)}

= z[X(z) − x(0)],

k=0

(12)

and the transform of the exponential function is Z{ak }

=

8

z . z−a

(13)

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Similarly, following the definition of Laplace transform L{f (t)} = F (s), the Laplace transform of the differentiation function is L{

df (t) } dt

sF (s) − f (0),

=

the transform of the time shift function is L{f (t − a)u(t − a)}

=

L{tR }

R! . sR+1

(15)

(16)

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=

f (t)e−st dt =

(14)

F (s)e−as ,

and the transform of the power function is

0

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R∞

Theorem 1. The solution of the hybrid system w(k+1, L) = w(k, L)−w(k, L− 1) − DL w(k, L − 1), where DL = ∂/∂L, with the w(0, L) = LR  initial  conditions  m k m P P k m (−1) R! and w(k, 0) = 0 is given by w(k, L) = (L − m)R−p . m p (R − p)! m=0 p=0

Proof.

Step 1. Z transform Defining Z{w(k, L)} =

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∞ P

w(k, L)z −k = W (z, L), the Z transform of Eq.

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k=0

(11) by using time difference property Eq. (12) is given by = W (z, L) − W (z, L − 1) − DL W (z, L − 1). (17)

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z[W (z, L) − w(0, L)]

Substituting the initial condition w(0, L) = LR into Eq. (17), we have the

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differentiated equation with the shifting term L − 1 as follows DL W (z, L − 1) = −W (z, L − 1) − (z − 1)W (z, L) + zLR with W (z, 0) = 0, (18)

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where the initial condition w(k, 0) = 0 has a Z transform Z{w(k, 0)} = W (z, 0) = 0.

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Step 2. Laplace transform By the definition L{W (z, L)} =

R∞ 0

W (z, L)e−sL dL = U (z, s) and shifting

property Eq. (15), we have L{W (z, L − 1)} = e−s U (z, s). We then apply Eqs. (14) and (15) to the differentiation and shifting property, respectively, to obtain 9

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e−s [sU (z, s) − W (z, 0)]

= −e−s U (z, s) − (z − 1)U (z, s) + z

Then, after some arrangement of Eq. (19), it follows U (z, s)

=

sR+1 (s

R!

. (19)

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L{DL W (z, L − 1)} = e−s [sU (z, s) − W (z, 0)]. On the other hand, L{LR } = R! is a simple result of Eq. (16). Thus, taking the Laplace transform on Eq. R+1 s (18), we can obtain sR+1

zes R! zR! = R+1 . (20) + 1 + (z − 1)es ) s [z − (1 − (s + 1)e−s )]

Step 3. Inverse Z transform

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transform on Eq. (20) leads to V (k, s)

=

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By defining Z −1 {U (z, s)} = V (k, s) and employing Eq. (13), the inverse Z R!

sR+1

(1 − (s + 1)e−s )k .

(21)

Referring to the binomial series, the term (1 − (s + 1)e−s )k in Eq. (21) can be represented as

  k (s + 1)m e−ms m m=0  X m   k X m p −ms m k s e . (−1) = m p=0 p m=0 k X

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(1 − (s + 1)e−s )k

(−1)m

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=

(22)

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Then, Eq. (21) can be rewritten as

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V (k, s)

=

=

R! sR+1 k X

(1 − (s + 1)e−s )k

(−1)m

m=0

 X m   k m R!e−ms (R − p)! . m p=0 p (R − p)! sR−p+1

(23)

Step 4. Inverse Laplace transform

Note that L−1 {V (k, s)} = w(k, L). Next, use Eq. (16) to obtain the power (R − p)! function L−1 { R−p+1 } = LR−p and Eq. (15) to result in the shifting property s −1 −ms (R − p)! } = (L − m)R−p . Then Eq. (23) can lead to L {e sR−p+1 k X m    X k m (−1)m R! w(k, L) = (L − m)R−p . (24) m p (R − p)! m=0 p=0

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Thus the proof is completed. Therefore, by using the particular differentiation operators Dm = ∂/∂m and p Dm = ∂ p /∂mp , there is an alternative expression for Eq. (24) as follows

w(k, L)

=

=

k  X m   X k m (−1)m R! (L − m)R−p m p (R − p)! m=0 p=0 k X

(−1)m

m=0

=

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 X   m k m p (−1)p Dm (L − m)R m p=0 p

  k (−1) (1 − Dm )m (L − m)R , m m=0 k X

m

(25)

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where the double summation becomes a single summation with differentiation operators.

w(k, L) , the probability Sk that LR at least k slots are “collision” as defined in Eq. (5) can be expressed as By Eq. (25) and the expression P (k, L) =

=

     k L k L 1 X (−1)q (1 − Dq )q (L − q)R . (26) P (k, L) = R k q k L q=0

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Sk

Therefore, in Eq. (6), the probability P[k] for the event that exact k slots

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are “collision” can now be expressed as       k+1 k+2 L P[k] = Sk − Sk+1 + Sk+2 − ... + (−1)L−k SL k k k  X    L v 1 X v v−k v q L = (−1) (−1) (1 − Dq )q (L − q)R LR k q=0 v q v=k

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=

    L v 1 XX v v q+v−k L (1 − Dq )q (L − q)R . (−1) LR v k q q=0

(27)

v=k

Note that probability P[k] , that there are exact k “collision” slots, contains a

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double summation with one differentiation operator in Eq. (27).

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We shall show in Lemma 1 that in Eq. (27), the double summation can be

reduced to a single summation. Lemma 1 will be further used in Theorem 2 to eliminate the differentiation operator Dq in Eq. (27).

Lemma 1. Consider the series f1 (q, k) =

L P

v=k

11

(−1)v−k

    L v v v k q

with

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210

Proof.

   L k k L−q

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0 ≤ q ≤ k. Then it can be simplified as f1 (q, k) = (−1)    L L−k k or f1 (q, L − k) = (−1) . L−k L−q

L−k

First, we will move the terms being irrelevant to the index v outside the summation so that f1 (q, k) can be simplified as f1 (q, k)

L X

=

(−1)v−k

v=k L−k X

=

(−1)v

L!(v + k)! (v + k − q)!q!(L − k − v)!v!k!

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v=0

    v L v q v k

L−k

X L! (L − k)!(v + k)! (−1)v (L − k)!k!q! v=0 (L − k − v)!v!(v + k − q)!     L−k (v + k)! L 1 X v L−k (−1) = . v k q! v=0 (v + k − q)!

=

(28)

Therefore, we will introduce the differentiation operator D x = d/dx  to cancel L−k P v L−k the variable q with the aid of two binomial series (−1) (x + 1)v = v v=0   k P k v (−x)L−k and x = (x + 1)k . Then, f1 (q, k) can be further expressed as v=0 v =

    L−k L 1 X v L−k (−1) Dxq (x + 1)v+k |x=0 v k q! v=0     L−k X L 1 q L−k Dx {(x + 1)k (−1)v (x + 1)v }|x=0 k q! v v=0   L 1 q D {(x + 1)k (−x)L−k }|x=0 k q! x   k   X L 1 q k v Dx {(−x)L−k x }|x=0 k q! v v=0   k   X L k L−k 1 (−1) Dxq {xL−k+v }|x=0 . k q! v=0 v

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f1 (q, k)

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It can be observed that Eq. (28) still contains variable q inside the summation.

AC

CE

=

= =

=

(29)

We note that 0 ≤ q ≤ k. Due to the substitution of x = 0 in the differentiation term Dxq {xL−k+v }, in Eq. (29), we only need to consider L − k + v = q and 12

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Dxq {xq } is a constant that is not affected by the substitution x = 0. Thus, the summation term f1 (q, k) can be further reduced as     L 1 k L−k f1 (q, k) = (−1) Dxq {xq } k q! q−L+k     L 1 k L−k = (−1) q! k q! q−L+k    L k = (−1)L−k . k L−q

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Thus the proof is completed.

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Finally, we can use Lemma 1 to reduce the differentiation operator in the

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probability P[k] will be stated in the following Theorem 2.

Theorem 2. For the event of exact k “collision” slots, its probability P[k] as given in Eq. (27) with one differentiation operator can be simplified as P[k] =   L−q   k X L − q  R! 1 L X q+k k (−1) q R−p without any differentiation LR k q=0 q p=0 (R − p)! p operator.

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225

Proof.

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230

Note that in Eq. (27), the lower  bound of the index v can be extended to v start from 0 as by the fact that = 0 for v < k, and thus Eq. (27) can be k rewritten as     L v L v 1 XX q+v−k v (−1) (1 − Dq )q (L − q)R . (31) P[k] = LR v=0 q=0 k v q

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Then exchange the order of the summation indices v and q in Eq. (31) to get P[k]

=

    L L 1 XX L v q+v−k v (−1) (1 − Dq )q (L − q)R . LR q=0 v=q k v q

(32)

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  v = 0 for v < k, the index v in the last k equation can start from k. Then, by using the summation term f1 (q, k) in

235

Once again by the fact that

Lemma 1, Eq. (32) can be rearranged as P[k]

=

    L L 1 XX L v q+v−k v (−1) (1 − Dq )q (L − q)R LR q=0 k v q v=k

13

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    L L X 1 X L v q q R v−k v (−1) (1 − Dq ) (L − q) (−1) LR q=0 q v k

=

v=k

=

By the result of Lemma 1, Eq. (33) can be reduced as =

   L k 1 X q q R L−k L (−1) (1 − Dq ) (L − q) (−1) . LR q=0 k L−q 

k L−q start from L − k so that, Due to the fact that

P[k]

=



(34)

= 0 for L − q > k, the index q in Eq. (34) can

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P[k]

(33)

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L 1 X (−1)q (1 − Dq )q (L − q)R f1 (q, k). R L q=0

    L k 1 L X L−q+k (−1) [1 − Dq ]q (L − q)R . L−q LR k q=L−k

(35)

By change of variable, q¯ = L − q or q = L − q¯, and its differentiation relationship Dq = −Dq¯, it follows that P[k]

=

  k   1 L X q¯+k k (−1) [1 + Dq¯]L−¯q (¯ q )R . LR k q¯=0 q¯

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240

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Furthermore, by the binomial series [1 + Dq¯]L−¯q =

L−¯ Pq p=0



(36)

 L − q¯ p R Dq¯ (¯ q ) and the p

R! q R−p , we can simplify the expression of P[k] as the equation = (R − p)! ECFF, which is shown in Eq. (37).

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Dqp¯ (¯ q )R

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P[k]

=

=

  L−¯   k Xq L − q¯ p 1 L X q¯+k k (−1) Dq¯ (¯ q )R q¯ p=0 p LR k q¯=0

  k   L−q X L − q  R! 1 L X q+k k (−1) q R−p . LR k q=0 q p=0 p (R − p)!

(37)

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Thus the proof is completed.

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3. NUMERICAL RESULTS We have derived two analytical results in Section 2.1 and Section 2.2 to evaluate collision probability in diverse multiple access communication systems 14

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with FSA protocol. In Section 2.1, the first formula Eq. (6) is derived by using the three-stage approach of GAA, and it is based on the probability P (k, L), 250

for 2 ≤ k ≤ L, in Eq. (4). In Section 2.2, the second formula Eq. (37),

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which is the main contribution of this paper, is an ECFF containing only two summations. These two analytical results will be compared and verified to be the same by evaluating the two mathematical formulae, Eqs. (6) and (37),

through numerical simulation by using Monte Carlo method. Under the FSA 255

protocol, we consider 109 frames for each run and the average of two runs is

presented. The values of the collision probability P[k] for the parameter L = 10,

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with two different settings R = 10 and R = 20, will be calculated and analysed

in the section. Simulation results and analytical results for the two settings are compared in Tables 3 and 4, respectively. We can see that the analytical and 260

simulation results match very well. The associated computational time increases as the number of R and/or L increase, and the difference of computational time between both methods becomes very significant, as listed in Table 5. For the

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L = R = 40 case, the computational time ratio of GAA/ECFF is 1,060,429 which is more than a million. The coding of GAA for the L = R = 50 case based on symbolic of Mathematica software becomes very huge and the computational

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265

time can be expected to be longer than one month. Therefore, for the cases L = R ∈ {50, 100, 200, 500, 1000}, we only implement the ECFF to find the

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probability distribution and the computational time.

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Table 3: Analytical and simulated P[k] for L = R = 10

k

0

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1 2 3 4 5

Monte Carlo (confidence level 95%) 3.62850 ×

10−4

3.69297 ×

10−1

1.14303 ×

10−1

± 8.51725 ×

10−7

± 2.15832 ×

10−5

± 1.42294 ×

10−5

5.53152 × 10−2 ± 1.02231 × 10−5 4.57864 × 10−1 ± 2.22811 × 10−5 2.85788 × 10−3 ± 2.38735 × 10−6

15

GAA 3.62880 ×

10−4

3.69299 ×

10−1

1.14307 ×

10−1

5.53123 × 10−2 4.57861 × 10−1 2.85768 × 10−3

ECFF 3.62880 × 10−4 5.53123 × 10−2 3.69299 × 10−1 4.57861 × 10−1 1.14307 × 10−1 2.85768 × 10−3

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Table 4: Analytical and simulated P[k] for L = 10 and R = 20 k

Monte Carlo (confidence level 95%)

0

0

1.25000 × 10−8 ± 5.00000 × 10−9

1.23320 × 10−8

1.23320 × 10−8

2.56555 × 10−3 ± 2.26228 × 10−6

2.56556 × 10−3

2.56556 × 10−3

2.83860 × 10−5 ± 2.38265 × 10−7

4

4.37099 ×

10−2

4.00318 ×

10−1

± 9.14323 ×

10−6

± 2.19118 ×

10−5

2.21999 × 10−1 ± 1.85858 × 10−5

5 6

2.66724 × 10−1 ± 1.97779 × 10−5

7

3.68006 ×

9

10−3

± 2.70796 ×

0.25

L=R=100

0.15

X: 54 Y: 0.1158

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4.00327 ×

10−1

2.21995 × 10−1 2.66720 × 10−1 6.09466 × 10−2 3.68262 ×

10−3

2.37588 × 10−5

50

100

2.83344 × 10−5 4.37107 × 10−2 2.21995 × 10−1 4.00327 × 10−1 2.66720 × 10−1 6.09466 × 10−2 3.68262 × 10−3 2.37588 × 10−5

ECFF L=R=50 ECFF L=R=100 ECFF L=R=200 ECFF L=R=500 ECFF L=R=1000 Monte Carlo L=R=50 Monte Carlo L=R=100 Monte Carlo L=R=200 Monte Carlo L=R=500 Monte Carlo L=R=1000

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X: 27 Y: 0.1612

L=R=200

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Probability of collision P[k]

0.2

4.37107 ×

10−2

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L=R=50

X: 14 Y: 0.2292

0 0

10−6

2.37915 × 10−5 ± 2.18133 × 10−7

10

2.83344 × 10−5

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6.09514 × 10−2 ± 1.06992 × 10−5

8

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0

1

3

0.05

ECFF

0

2

0.1

GAA

X: 133 Y: 0.07336

L=R=500 X: 265 Y: 0.05185

150

L=R=1000

200 250 300 350 Number of "collision" slots

400

450

Figure 1: Collision Probability P[k] for L = R ∈ {50, 100, 200, 500, 100}

16

500

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Table 5: Computational time GAA (Sec.)

ECFF (Sec.)

L = R = 10

0.0156

0.0156

L = 10, R = 20

0.7644

0.0156

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Measurement

0.8892

0.0156

181.975

0.0156

L = R = 30

183.566

0.0156

L = R = 40

33085.4

0.0312

L = R = 50

N/A

0.0936

L = R = 100

N/A

0.7176

L = R = 200

N/A

7.2853

N/A

193.05

N/A

3146.5

L = R = 500 L = R = 1000

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L = R = 20 L = 20, R = 30

To compute the collision probability via Eq. (6) based on GAA, a (k − 1)−

270

tuple summation should be carried out for P (k, L) in Eq. (4) so that there are

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too many repeated summations, each one containing a large number of indices, should be calculated, especially for large L. It notes that the proposed ECFF can significantly reduce complicate calculations and become evaluable when the parameters grow larger. For example, using the ECFF, we can compute P[k]

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275

for L = R ∈ {50, 100, 200, 500, 1000} as shown in Figure 1. Simulation result

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displayed in Figure 1 further validates the correctness of the ECFF.

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4. CONCLUSION AND DISCUSSION Analysis of collision probability is important for performance evaluation of

280

diverse multiple access communication systems based on FSA protocol. How-

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ever, it is difficult to derive an exact and closed-form solution for collision probability under general settings. Recently, an exact solution, without approximation, based on the GAA [31] has been proposed, yet the numerical computation will become difficult when the number of slots becomes larger. Especially, for

285

L ≥ 16, the computation burden in GAA is usually over the capacity limit of normal computing facilities to result in unacceptable round-off errors. In this 17

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paper, we have developed an ECFF for the collision probability that can not only overcome the computational deficiency of the GAA in the presence of a large number of slots, but also reduce the computation complexity of collision probability.

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290

The ECFF is derived by three stages in order to compute the final collision probability. At the first stage where the event that a set of k particular slots are “collision” is considered, a recursive equation containing three inde-

pendent variables is derived for the probability of this event. By introducing 295

a differentiation operator, one independent variable is omitted in the recursive

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equation and simultaneously the recursive equation becomes a reduced hybrid

equation containing a continuous-time variable and a discrete-time one. Then, with the aid of Laplace transform, Z transform, and algebraic properties of binomial coefficients and power series, at the third stage where the event that 300

exact k slots are “collision” is considered, the probability of this event can be originally expressed by using a triple summation with a differentiation operator.

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Finally, we have shown that the above triple-summation solution can be simplified as a compact double-summation type. Verification of the proposed algo-

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rithm and comparison with the GAA have also been made by simulation study. REFERENCES

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