Proposed Multiple Queue Finite State Markov Chain System Model

To model the communication system, we combine the FSMC at physical layer with the multiple queues’ status at MAC layer to construct a finite state Markov chain. Station- ary distribution of the obtained Markov chain is calculated to analyze the diverse system performance, in terms of packet loss rate and average system delay.

3.3.1

Queue Behavior Modeling

We consider the M/M/1/B_i/FCFS model forQ_i in the proposed system, that is a queueing system where customers arrival is modeled by Poisson process and the exponentially dis- tributed service times from the server are requested. Moreover, the system has only one server, an finite queue length (Bi) and the customers are served on a FCFS basis.

Let Ai_t (i ∈ [1,M]) denote the number of packets arriving at time frame t for Qi, which is assumed to be Poisson distributed with parametersλiT_f,

P(Ai_t =a)=            (λiT_f)aexp(−λiT_f) a! , i=1,2, ...,M and ifa≥0 0, otherwise , (3.12)

whereE[Ai_t]=λiT_f withE[Ai_t]∈A:={0,1, . . . ,∞}andE[·] denoting the expectation of a random variable.

frametforQi. Obviously,Ri_t should satisfy the following constraints, 0≤Ri_t ≤Q_ti₋₁, i∈[1,M] M X i=1 Ri_t ≤Rmax(kt), (3.13)

whereQi_t−1 is the number of packets inQiat the end of time framet−1 with the definition of Qi_t−1 ∈ Qi := [0,Bi] and Rmax(kt) denotes the maximum allowable number of packets

transmitted per time frame at physical layer given channel state Sk at time framet. As-

suming that np packets can be accommodated during each time frame wherenp depends

on the designer’s choice, we have Rmax(kt) = nP ×R(kt) where R(kt) is the symbol rate

corresponding to channel stateS_k. Therefore, we can get the recursions of the queue state forQias

Qi_t =min(Bi,max(0,Qi_t−1−R

i

t)+Ait), i= 1,2, ...,M (3.14)

where Ai_t is the packets arrived at time framet for Qi, denoting the network traffic con- dition and obviously, it is independent with other system parameters. Therefore, we can isolateAi_twhen considering the system state transition and construct the finite state Markov chain system model with state pair (Kt,Q1_t−1,Q2_t−1, ...,Qi_t−1, ...,Q_tM₋₁) to analyze the system

behavior by combining (3.12) to (3.14).

3.3.2

Priority-Based Rate Allocation

As discussed earlier, the queue state recursion relies on the queue status at both current and previous time frame as well as the physical layer transmission data rate corresponding to current channel state. Therefore, the problem comes out to be how to allocate the data rate at physical layer to the multiple MAC layer queues according to their diverse QoS requirements.

To resolve this problem, we propose a priority-based rate allocation (PRA) algorithm, where multiple queues are assumed to have different priorities and the physical layer data rate is distributed to the multiple MAC layer queues proportionally with higher priority queue getting higher data rata. Mathematically, the relations between the data rates of different queues are defined as:

R_tM =βRk_t−1 = β2R_tk−2 = . . .= βM−1R1_t M X i=1 Ri_t ≤ Rmax(kt) (3.15)

whereβ is called asproportional weight, adjusting which the data rate allocation process could be controlled. Combining (3.13) with (3.15), we get the constraints for the data rates of the multiple queues as

R1_t(1−βM) 1−β ≤ R max₍ kt) R_tM =βRk_t−1 = β2R_tk−2 = . . .= βM−1R1_t 0≤Ri_t ≤ Q_ti₋₁, i∈[1,M]. (3.16)

To explain the PRA algorithm in detail, let A denote the set of the queues which are waiting for data rate allocation, where initially we set A = {Q1,Q2, . . . ,QM}. To maximally utilize the physical layer resources while satisfying the constraints in (3.16), a iterative-based PRA implementation is shown as

Step a: Initialization: SetA = {Q1,Q2, . . . ,QM}andRmax(kt) according to current

Step b: ForQi ∈A, calculate (Ri_t)0 according to the following equation: X Qi∈A (Ri_t)0 =Rmax(kt) (Ri_t)0= qM−i(R1_t)0 (3.17)

Step c: If∀Qi ∈ A, (Ri_t)0 ≤ Q_ti₋₁, setRi_t = (Ri_t)0 and then, terminate the algorithm; otherwise, go to Step d.

Step d: FindQi ∈A, where (Ri_t)0 ≥ Qi_t−1, setRi_t = Qi_t−1, A=A -{Qi}andRmax(kt)=

Rmax(kt)−Ri_t;

Step e: If A==φ, terminate the algorithm; Otherwise, go to Step b.

3.3.3

System Modeling Using Finite State Markov Chain

Given the state pair (Kt,Q1_t−1, . . . ,Qi_t−1, . . . ,Q_tM₋₁) derived above, we now formulate the

state transition probability matrixPtwith its elementsP((k,q₁,...,q_M),(k0,q0

1,...,q 0

M))which de-

notes the state transition probability from system state (Kt,Q1_t−1, . . . ,Q_tM₋₁) to system state

(K_t+1,Q1_t, . . . ,Q_tM). Mathematically, the state transition probability can be computed by

P_((k,q

1,...,qM),(k0,q₁0,...,q0_K)) = Pk,k0·[P((q1,...,qM),(q₁0,...,q0_M))|K=k0], (3.18)

whereP_k,k0 is the channel state transition probability inP_cand P_((q

1,...,qK),(q0₁,...,q0_K))|K=k0 can be calculated as P_((q 1,...,qM),(q0₁,...,q0_M))|K=k0 = M Y i=1 P(Qi_t =q0_i|Qi_t−1 =q_i,Ri_t =r_i), (3.19)

whererirepresents the data rate allocated to queueQi. Using (3.12) and (3.14), we compute P(Qi_t =q0_i|Qi_t−1 = qi,Ri_t = ri) for Qias P(Qi_t =q0_i|Qi_t−1 = qi,Rit = ci)=                P(Ai_t =q0_i −max{0,qi−ri}), if 0≤ q0_i < Bi 1− P 0≤q0 i<Bi P(Qi_t =q_i0|Qi_t−1 = q1,Ri_t = ri), ifq0_i = Bi. (3.20)

3.3.3.2 Stationary distribution of Markov chain

From the state transition matrix Pt, we calculate the stationary distribution of the Markov

chain with definition:

Π(k,q₁, . . . ,q_M)= P(K = k,Q1 =q₁, . . . ,QM =q_M) := lim t→∞P(Kt =k,Q 1 t−1 =q1, . . . ,Q M t−1 = qM). (3.21)

We define the stationary distribution row vector as

Π= [Π(0, . . . ,0), . . . ,Π(k,q1, . . . ,qM), . . . ,Π(K,B1, . . . ,BM)]. (3.22)

The stationary distribution of state pair (Kt,Q1_t−1, . . . ,Qi_t−1, . . . ,QM_t−1) can be finally

obtained by solving the following equations

Π = ΠPsys, and

X

Π(k,q₁, . . . ,q_M)= 1. (3.23)