Joint Markov Chain Analysis

2.4 Conclusion and Future Work

3.1.4 Performance Analysis

3.1.4.3 Joint Markov Chain Analysis

CW_max^L = 1024, and K = 1, the total number of transmit states in the L node’s MC becomes 7,040 and thus|SL|>10,000. Therefore, it is desirable to reduce|SL|to mitigate the complexity in analyzing the joint MC. To do so, we introduce a simplified MC that selectively groups the states of the L node’s MC in Fig. 10, which is shown in Fig. 13. Specifically, the transmit states(i,0^j),∀iin Fig. 10 are combined into the single statej in Fig. 13, and the backoff states (i, k),∀kin Fig. 10 are combined into the single state(i+M+1)in Fig. 13. Then, the stationary distribution of the simplified MC can be determined by using the stationary distribution of the original MC in Section 3.1.4.1.

Unless otherwise specified, Lt will henceforth imply the state of the L node’s simplified MC at slot t.

3.1.4.3.1 Transition Matrix of H_t in RJMC As introduced earlier, the JMC is con- sisting of the outer MCLt and the inner MCHt. The transition matrixP_H^l,l0 of the inner MC is consisting of the transition probabilities P_H^l,l0(h, h⁰) := P(H_t+1 =h⁰|L_t+1 =l⁰, L_t =l, H_t = h), as shown in Eq. (18). P_H^l,l0(h, h⁰) is derived from the MC in Fig. 11, where the two parameters

p^H_c andp^H_b are set differently according to the state Lt of the outer MC.

P_H^l,l0=

















P_H^l,l0(1,1) P_H^l,l0(1,2) · · · P_H^l,l0(1,|SH|) P_H^l,l0(2,1) P_H^l,l0(2,2) · · · P_H^l,l0(2,|SH|)

... ... . .. ...

P_H^l,l0(|SH|,1) P_H^l,l0(|SH|,2) · · · P_H^l,l0(|SH|,|SH|)

















(18)

In the asymmetric coexistence, an H node in the backoff stage decreases its BC at every slot when other(n_H−1)H nodes are also in the backoff stage since it does not recognize L node’s signal, and a packet is transmitted once the BC hits zero. An H node’s transmission, however, has different consequences according to the type of periods it is transmitting in. In an MCOT period, the transmission will always be in collision due to the ongoing transmission by the L node. The overlapping transmission that starts in an MCOT period is also in collision since it partially overlaps with an MCOT period. In an OW period, a transmission of an H node will be in collision when other H nodes concurrently start transmissions. Therefore, we define the collision and busy probabilities of an H node for MCOT and OW periods as follows:

• p^H,Oc ,p^H,Mc : the probability that an H node’s packet transmission is in collision, where the transmission starts and completesin an OW periodand in an MC period, respectively.

• p^H,O_b ,p^H,M_b : the probability that an H node senses the channel busyin an OW period and in an MC period, respectively.

Based on the above definitions, p^H_c and p^H_b are set as p^H,M_c andp^H,M_b for an MCOT period, andp^H,O_b andp^H,Oc for an OW period. To capture the overlapping transmission,1A(Lt, Lt+1) is set to one whenL_t =M and L_t+1 ∈[M + 1, M +m_L+K+ 1] while it is set to zero for other transitions. Thanks to the concept of the two periods and Eq. (17), we only need to consider three transition matrices for the inner MC as follows:

• P_OW is P_H^l,l0 withp^H,Oc ,p^H,O_b , and 1_A(l, l⁰) = 0, which is used whenl, l⁰∈[M + 1, M +m_L+K+ 1].

• P_OL is P_H^l,l0 withp^H,Mc ,p^H,M_b , and 1_A(l, l⁰) = 1, which is used whenl=M,l⁰∈[M + 1, M +mL+K+ 1].

• PMC is P_H^l,l0 withp^H,Mc ,p^H,M_b , and 1A(l, l⁰) = 0, which is used whenl, l⁰∈[1, M].

3.1.4.3.2 Transition Probability of L_t in RJMC According to Eq. (9), the proba- bility P(Lt+1|L_t, Ht) is multiplied to the row of the inner MC’s transition matrix (i.e., Eq. (18)) that corresponds toH_t. This relationship can be expressed by exploiting a diagonal matrixD^l,l0

which consists ofD^l,l0(h) :=P(Lt+1=l⁰|L_t=l, Ht=h), as shown in Eq. (19).

D^l,l0=





















D^l,l0(1) 0 · · · 0 0 0 D^l,l0(2) · · · 0 0

... ... . .. ... ...

0 0 · · · D^l,l0(|SH| −1) 0 0 0 · · · 0 D^l,l0(|SH|)





















(19)

The diagonal matrix is determined by considering the following five transition cases of the outer MC ¹¹, such as

• Case 1: Transition from a state in the MCOT stage of Fig. 13 to the next state in the MCOT stage, i.e.,L_t∈ {1, . . . , M −1} and L_t+1 =L_t+ 1.

• Case 2: Transitionfromstate M tostate 1, i.e., Lt=M and Lt+1= 1.

• Case 3: Transition from state M to a state in the backoff stage of Fig. 13, i.e., L_t =M and Lt+1 ∈ {M + 1, . . . , M +mL+K+ 1}.

• Case 4: Transitionfroma state in the backoff stagetostate 1, i.e.,L_t∈ {M+ 1, . . . , M+ mL+K+ 1} andLt+1= 1.

• Case 5: Transition from a state in the backoff stage to itself, i.e., Lt = Lt+1 ∈ {M + 1, . . . , M +m_L+K+ 1}.

Case 1, 2, and 3 correspond to the situation where the L node is in transmission, during which the L node is not affected by the state of H nodes because the L node keeps transmitting during an MCOT period once it has been started. Therefore, all the elements of D^l,l0 have the same value regardless of H_t such that D^l,l0(h) = P_l,l⁰,∀h whereP_l,l⁰ is a constant for given (l, l⁰). In Case 4 and 5, transitions are initiated when the L node is in backoff states. Since the operation of the L node such asdeferorfreezeis affected by the state of an H node, the elements ofD^l,l0 should be derived according toHt. In addition, Cases 4 and 5 need to reflect the effect of other(n_H−1)H nodes. By the law of total probability,D^l,l0(h) :=P(L_t+1=l⁰|L_t=l, H_t=h) can be rewritten as follows:

P(L_t+1=l⁰|L_t=l, H_t=h) = X

h2∈SH

... X

h_nH∈SH

h

P(H_t²=h2, ... , H_t^nH=hnH|L_t=l, Ht=h)· P(Lt+1=l⁰|L_t=l, Ht=h, H_t²=h2, ..., H_t^nH=hnH)

i

. (20)

Therefore, we can consider the behaviors of other(n_H −1)H nodes in derivingD^l,l0. Then,D^l,l0 is determined as follows.

11In the other transitions not considered in the five transition cases, all the elements ofD^l,l0 have zero since such transitions are impossible.

Case 1: In this case,D^l,l0(h) =Pl,l⁰,∀handPl,l⁰ = 1since the transition occurs with probability 1 within an MCOT.

Case 2: In this case, D^l,l0(h) = P_l,l⁰,∀h and P_l,l⁰ can be obtained by summing all the possible transitions from(a,0^M) to(b,0¹) wherea, b∈[0, m_L+K]in the L node’s MC. Therefore,

P_l,l⁰=







mL+K−1

X

i=0

p^s_i(1−p_d) +p^s_mL+K





 1 W₀ +

mL+K−1

X

i=0

p^s_i p_d 1

W_i+1, (21)

wherep^s_i =b^M_i,0/PmL+K

a=0 b^M_a,0 is the probability that the L node is in backoff stage iat theM-th MCOT slot.

Case 3: In this case, D^l,l0(h) = P_l,l⁰,∀h. When switching into state M + 1 in the backoff stage, P_l,l⁰ is obtained by summing all the transitions from (i,0^M) to (0, k), i ∈ [0, m_L+K], k∈[1, W0−1], in the L node’s MC. When switching into stateM+ 1 +i,i∈[1, mL+K], in the backoff stage, P_l,l⁰ can be derived by considering all the transitions from (a−1,0^M) to (a, k), a∈[1, m_L+K]k∈[1, W_i−1], in the L node’s MC. Therefore,

P_l,l⁰=







nPmL+K−1

i=0 p^s_i(1−p_d) +p^s_m

L+K

oW0−1

W0 , η(l⁰) = 0, p^s_η(l0−1)·pd·^W_W^η(l0)−1

η(l0) , otherwise,

(22)

whereη(l) =l−M −1, M+ 1 ≤l ≤M+m_L+K+ 1, is the backoff stage corresponding to state lin the L node’s MC.

Case 4: In this case,lis a state in the backoff stage andl⁰= 1. When an H node is transmitting while the L node is in the backoff stage, the L node should freeze the backoff procedure, i.e., it stays at the backoff stage. Therefore, we have D^l,l0(h) = 0 whenh corresponds to a state of an H node’s transmit stage. On the other hand, when the H node is in its backoff stage, the L node can start transmission (i.e., it switches into state 1) if all other (nH−1) H nodes are also in their backoff stage, and then the L node’s BC becomes zero after decrement. Accordingly, we can obtainD^l,l0(h)by using Eq. (20). The first term in Eq. (20) is the probability that in statel of the L node’s MC, the H node senses the channel idle sinceh, h₂, ..., h_nH ∈S_BO^H whereS_BO^H is the set including all the backoff states in the H node’s MC. The second term is the probability that the L node’s BC becomes zero after decrement, which is the same as the probability that the L node’s BC happens to be one at the current state. As a result,D^l,l0(h) is derived as

D^l,l0(h) =











0, h∈S_{T X}^H ,

(1−p^H,l_b )· ^bη(l),1

PWη(l)−1 k=1 bη(l),k

, otherwise, (23)

whereS_{T X}^H is the set including all the transmission states in the H node’s MC, and p^H,l_b is the probability that in statel of the L node’s MC, the H node senses the channel busy.

Case 5: In this case,lis a state in the backoff stage andl⁰=l. When an H node is transmitting while the L node is in the backoff stage, the L node should freeze the backoff procedure and thus

D^l,l0(h) = 1wherehcorresponds to an H node’s transmit stage. When both the H node and the L node are in their backoff stage, the L node will stay in the same backoff state if at least one of (n_H−1)H nodes is in transmission or if the L node’s BC does not become zero after decrement when all (nH−1)H nodes are in their backoff stage. As a result, Case 5 is the opposite of Case 4, and thus we can obtainD^l,l0 of Case 5 by subtracting Eq. (23) from 1. Hence, we have

D^l,l0(h) =











1, h∈S_{T X}^H ,

1−(1−p^H,l_b )· ^bη(l),1

PWη(l)−1 k=1 bη(l),k

,otherwise. (24)

3.1.4.3.3 Evolution of RJMC By using the equations in Sections 3.1.4.3.1 and 3.1.4.3.2, the transition matrixP^Z of the RJMC can be constructed as in Eq. (25) and in [38]. The evo- lution of the RJMC is given as π_t+1=π_t·P^Z where

πt = [πt(1,1), . . . , πt(M+m_L+K+ 1,|SH|)], (26)

πt(l, h) = P(Lt=l, Ht=h). (27)

The marginal distributionπ_t^l is defined as

π_t^l= [πt(l,1), . . . , πt(l,|SH|)]. (28) Finally, the probability that an H node is transmittingper slot, denoted byτ_H, can be obtained as

τ_H =

M+mL+K+1

X

l=1

π^l_t·m⁰_T (29)

wherem⁰_T is a column vector consisting of 0’s and 1’s that filters out H node’s non-transmitting states.

By using π_t,τ_L can be derived as

τL=

M

X

l=1

π_t^l

1, (30)

wherek k₁ is the 1-norm.

3.1.4.3.3.1 Collision and Busy Probabilities of an H node in a State of L node’s MCCollision and Busy Probabilities of an H node in a State of L node’s MC In the analysis introduced so far, we have not derived p^H,l_b in Eqs. (23) and (24) when deriving D^l,l0 of Cases 4 and 5, and p^H,O_b ,p^H,Oc , and p^H,M_b in Section 3.1.4.3.1. First, by applying the law of total probability, p^H,O_b ,

Lt+1 Lt

12···M−1MM+1···M+mL+K+1 1

                       

01·PMC···000···0 00···000···0 . . . . . . . .. . . . . . . . . . . .. . . .

00···1·PMC00···0 00···01·PMC0···0 PM,1·PMC0···00PM,M+1·POL···PM,M+mL+K+1·POL DM+1,1 ·POW0···00DM+1,M+1 ·POW···0 DM+2,1 ·POW0···000···0 . . . . . . . .. . . . . . . . . . . .. . . .

DM+mL+K+1,1 ·POW0···000···DM+mL+K+1,M+mL+K+1 ·POW

                       

2 . . .

M−2 M−1 M M+1 M+2

. . .

M+mL+K+1

(25)

p^H,Oc , and p^H,M_b can be derived as

p^H,O_b =

M+mL+K+1

X

l=M+1

p^H,l_b ·

π^l_t

1

PM+mL+K+1 q=M+1

π_t^q 1

, (31)

p^H,O_c =

M+mL+K+1

X

l=M+1

p^H,l_c ·

π^l_t

₁ PM+mL+K+1

q=M+1

π_t^q

1

, (32)

p^H,M_b =

M

X

l=1

p^H,l_b · π_t^l

1

PM q=1

π^q_t 1

. (33)

wherep^H,lc is the probability that an H node experiences collision in statel of the L node’s MC.

Then, to derivep^H,l_b andp^H,l_c , we adopt the method introduced in [31] which has derived busy and collision probabilities for the Wi-Fi nodes exposed to each other while modeling a Wi-Fi node via MC. The method proposed a new concept of ‘network Markov chain’ that reflects the interaction amongnWi-Fi nodes, whose state distribution is defined as

p_act= [p_act(0), p_act(1), . . . , p_act(n)] (34) wherep_act(i) is the probability thatiWi-Fi nodes are in transmission. In [31], p_act is obtained by using individual state distributions of the nWi-Fi nodes. Since the network Markov chain includes all possible interactions among Wi-Fi nodes, pact is exploited to derive the busy and collision probabilities of a Wi-Fi node.

Similar to [31], our JMC also models an H node’s behavior using MC and assumes that H nodes are exposed to each other, given statel of the L node’s MC. Therefore, we can apply the network Markov chain to capture the interaction amongnH H nodes, given statel. In our JMC, the state distribution of an H node’s MC given statel can be obtained usingπ^l_tin Eq. (28). As in [31], by utilizing π^l_t, we can derive the state distribution p_H of the network Markov chain amongnH H nodes given statel, wherep^l_H is defined as

p^l_H = [p^l_H(0), p^l_H(1), . . . , p^l_H(n_H)] (35) and p^l_H(i) is the probability that iH nodes are in transmission given state l. Finally, p^H,l_b and p^H,lc can be derived following the procedure in [31].

In addition, we can derive the probabilityp^L,l_b that the L node senses the channel busy while in state lasp^L,l_b = 1−p^l_H(0). Therefore, by the law of total probability,p^L_b can be obtained as

p^L_b =

M+mL+K+1

X

l=M+1

p^L,l_b ·

π_t^l

1

PM+mL+K+1 q=M+1

π^q_t 1

. (36)

3.1.4.3.3.2 Collision Probability of an H NodeCollision Probability of an H Node Fig. 12 describes a cycle consisting of an MCOT period and an OW period, where the MCOT period has a duration ofM and the OW period has a random durationD_L, both in slots. On average,

there exist x packets within an MCOT period and y+w packets within an OW period, and there existzpackets that stretch from an MCOT period to the following OW period. Note that z <1 since there can be at most one overlapping packet per MCOT and not every MCOT will experience such a packet, and we denote the portion of z in the MCOT period side by z1 and that in the OW period side byz₂, wherez₁+z₂ = 1.

p^H_c denotes the probability that a packet of an H node is in collision, and p^H,Oc is the probability that an H node’s packet is in collision when it starts and completes its transmission in an OW period. Accordingly, both collision probabilities are measured per packet. Using w, x,y, andz, we can expressp^H_c and p^H,Oc as

p^H_c = x+y+z

x+y+z+w, p^H,O_c = y

y+w. (37)

Eq. (37) can be rewritten as

p^H_c = x+z1z

x+y+z+w + z2z

x+y+z+w + (y+w)p^H,Oc

x+y+z+w. (38)

In Eq. (38), the three terms on the right hand side represent the conditional probabilities that an H node is in transmission 1) during an MCOT period, 2) during an OW period for an overlapping transmission, and 3) during an OW period for a non-overlapping transmission, which are derived as

x+z1z

x+y+z+w =

PM

l=1π^l_t·m⁰_T PM+m_L+K+1

l=1 π_t^l·m⁰_T, (39)

z2z

x+y+z+w =

PM+mL+K+1

l=M+1 π_t^l·m⁰_V PM+m_L+K+1

l=1 π^l_t·m⁰_T, (40)

y+w

x+y+z+w =

PM+mL+K+1

l=M+1 π_t^l·(m⁰_T −m⁰_V) PM+m_L+K+1

l=1 π_t^l·m⁰_T , (41)

wherem⁰_V is a column vector consisting of 0’s and 1’s that filters out the H node’s non-overlapping states.

3.1.4.3.3.3 Doubling Probability of the L NodeDoubling Probability of the L Node Doubling of the L node’s CW is triggered when the RSF is in collision, where the collision implies that at least one slot of the subframe is in collision due to concurrently transmitting H nodes. Since H nodes always decrements its BC at every slot when no H node is in transmission, the doubling probability p_d becomes the probability that at least one H node has its BC smaller than the length of a subframe (in slots) at the start of the RSF, such as

p_d=C_sf(r), when RSF = r, (42)

Csf(r) = 1− (

X

{h|SFslot<BC(h)}

π^(r−1)·SF_t ^slot+1(h)

π_t^{(r−1)·SFslot+1} 1

)nH

, (43)

where BC(h) is the BC value corresponding to state h of an H node’s MC, π_t^l is the marginal distribution as previously defined in Section 3.1.4.3.3, and SF_slot is the length of a subframe in slots. Note thatC_sf(r)will be called thesubframe collision probability for subframer.

3.1.4.4 Throughput and Access Delay Analysis In this subsection, the throughput and