Analytical Model for Conventional Greedy Scheme

The analytical model for the conventional GS scheme is constructed by adopting the Markov chain techniques. Based on the Markovian approach, there exists a state transition diagram consisting of internal states and the respective state transition probability. In this subsection,

Sequence Number

Figure 5.2: The three critical parameters for the internal state definition of the GS scheme:

(a) the transmitter correctness bitmap, (b) the middle union, and (c) the state index.

the internal states for the analytical model of the GS scheme will first be defined. The underlying state transition diagram with the transition probability for the corresponding state transition is also properly formulated. Finally, the derivation of window utilization for the GS scheme concludes the analytical model. It is noted that all the context will be provided based on the generic analytical model with window size W . An illustrative example under W = 3 will be addressed for further explanation.

5.3.1.1 Internal States

In order to define the internal states of GS scheme, three parameters are first introduced, including the transmitter correctness bitmap, the middle union, and the state index. The transmitter correctness bitmap represents the Boolean correctness array for all numbered data packets from the transmitter’s perspective. The state index is a set of W binary digits with its most significant bit located at the first zero of the transmitter correctness bitmap.

The middle union is acquired by implementing the bitwise OR operation on the current state

S

Figure 5.3: State transition diagram for the conventional GS scheme under the window size W = 3: each state transition is represented as a unidirectional link with the causal middle union of width W .

index and the feedback block ACK bitmap. It is also noted that the state index will be utilized to define its respective internal state.

The illustrative example for the three critical parameters under the window size W = 3 is shown in Fig. 5.2. Initially, the transmitter correctness bitmap is an all-zero array, which results in the state index k = [000]. The transmitter starts to deliver the numbered data packets D1, D2, and D3 to the receiver. It is assumed that a feedback block ACK packet with a bitmap of b = [101] is consequently acquired by the transmitter. As a result, the middle union can be computed as m = k ∪ b = [000] ∪ [101] = [101]. As illustrated in Fig.

5.2, the transmitter correctness bitmap is also updated by recording the bit values in the middle union into its corresponding bits. As the transmitter correctness bitmap is changed, the state index is consequently shifted to become k = [010] based on its definition. According to the conventional GS scheme, the transmitter redelivers the previously corrupted packet D2 together with the newly scheduled packets D4 and D5, and receives a feedback block ACK packet consisting of a bitmap b = [100]. As a consequence, the new state index will be calculated and obtained as k = [000] as in Fig. 5.2.

Based on the definition, the state index must have the leading zero to represent either a previously corrupted or a newly scheduled packet. Therefore, the conventional GS scheme will possess a total of 2^{W −1} internal states that are denoted as S_k with the state index k ∈ [0, 1, ..., (2^{W −1} − 1)]. As shown in Fig. 5.3, four states S₀ to S₃ exist in the state transition diagram for the conventional GS scheme under the window size W = 3. In order to facilitate the explanation, each state can be represented by either the decimal or the binary subscript, i.e., S₀ ≡ S_[000], S₁ ≡ S_[001], S₂ ≡ S_[010], and S₃≡ S_[011].

5.3.1.2 State Transition Diagram

In the state transition diagram of window size W , there exist 2^{W −1} internal states and their corresponding state transitions. Each transition between two states is represented as a uni-directional link from the original state S_i to the resulting state S_k. Moreover, the potential middle unions that result in the state transition are also shown around the link. The rep-resentative state transition diagram under the window size W = 3 is illustrated in Fig. 5.3.

Considering that the transition from S_[001] to S_[010] results from the middle union m = [101], the middle union m = [101] will consequently let the transmitter increase the SSN by one, causing the state index to become k = [010], i.e., the state of S_[010]. It is noted that the mechanism for acquiring the resulting state can be generalized by the rule as to remove all the leading ones in the middle union and pad an equal number of zeros from the right side.

5.3.1.3 State Transition Probability

As shown in Fig. 5.3, the state transition probability from S_[001] to S_[010] can be computed from the bit values within the middle union m = [101] as G[001]→[010]= (1 − p_e) · p_e, where p_e denotes the packet error probability and G[001]→[010]represents the transition gain from S_[001]

to S_[010]. It is noticed that the rightmost bit within the middle union is not considered in the computation of packet error probability since the data packet was correctly received in the previous transmission. The generic transition probability from S_i to S_k under the window size W is modeled as the transition gain G_i→k which can be computed via Algorithm 5.

As shown in the for loop of Algorithm 5, the transition gain G_i→k is acquired by adopting

Algorithm 5: Transition Gain for Greedy Scheme Data: i, k, and r are unsigned binary integers of W digits Result: Gi→k

shift r one bit left with zero padding

9

the exhaustive search method, i.e., to scan from every possible middle union m. Furthermore, the bitwise AND operation of the possible middle union and the state index k will consistently be equal to k since the bit values that denote correctly received packets in the transmitter correctness bitmap will always have the one value. For example, it can be observed that the bitwise AND operation on the middle union m = [101] and the original state index k = [001]

is still equivalent to the original state index k = [001].

As a result, the algorithm employs the function bitwise and() and the state index i as a mask to filter out those unsuitable middle unions m recognized by the discriminant that the bitwise AND operation of i and m is not equal to i. Moreover, the number of newly correctly acknowledged packets is represented as the variable α. The variable β records the number of the remaining zeros, i.e., the number of those unacknowledged packets. The two variables α and β are utilized for the computation of packet error probability based on the corresponding middle union m. For the next step, the while loop generates the resulting state index r by the rule to suppress all leading one of the middle union m and to pad an equal number of zeros from the right-hand side. Consequently, the transition gain G_i→k can be computed via the sum of error probability from each suitable middle union m leading to the resulting state index r = k.

5.3.1.4 Window Utilization

The window utilization for the GS scheme can be derived from the saturated probability of internal states after the occurrence of an infinite number of state transitions. The saturated probability Θ_k for each internal state S_k under the window size W can be computed as the sum of all the possible 2^{W −1} state transitions as

Θ_k=P₂W −1−1

i=0 G_i→kΘ_i, (5.1)

for ∀k ∈ [0, 1, ..., 2^{W −1}− 1]. However, these state equations form an underdetermined linear system which has an infinite number of solutions for all the Θ_k. Therefore, for solving the saturated probability Θ_k, an additional constraint is imposed in order to make the underde-termined system possess a finite number of solutions, i.e.,

2^{W −1}X−1 k=0

Θ_k= 1 (5.2)

since the sum of all saturated state probability Θ_kin a Markov state transition diagram should be equal to one. After the derivation of saturated probability Θ_k from (5.1) and (5.2), the window utilization U_GS^W of the GS scheme under the window size W can be obtained as

U_GS^W =

P₂W −1−1

k=0 (1 − p_e)Z_W(k)Θ_k

W , (5.3)

where the subscript GS is adopted to denote the conventional GS scheme. Z_W(k) represents the number of zeros within the state index k of W digits, i.e., to indicate the number of unacknowledged data packets. It is noted that the product (1 − p_e) · Z_W(k) denotes the expected total number of properly acknowledged data packets that is forecasted in the next state transition of S_k.

An illustrative example under the window size W = 3 is provided as follows. Based on

(5.1), the four state equations under W = 3 are represented via the matrix form as follows: result, the window utilization for the W = 3 case becomes

U_GS³ = (1 − p_e)(3Θ₀+ 2Θ₁+ 2Θ₂+ Θ₃)

3 , (5.9)

which concludes the description and derivation of the analytical model for conventional GS scheme.