Derivation of the Parameters for Queuing Network

5.2 Queuing Analysis of IEEE 802.11s for a Specific Rate Region

5.2.2 Derivation of the Parameters for Queuing Network

Two different events of packet arrival can occur in a mesh STA. First, the packet may arrive from the clients, and second the packet may be a relayed packet. Let us assume that Λ_edenotes the packet arrival rate from the clients to every mesh STA. Thevisit ratio of a mesh STA, is defined as the average number of time a packet is forwarded by the mesh STA. The visit ratio ofST A_i, denoted as e_i, is given by,

e_i=p_0i+ X

r⁰∈R

X

ST Aj∈N₁ⁱ(r⁰)

p_ji(r⁰).e_j (5.1)

where p0i denotes the probability that a packet enters the queuing network from the clients ofST A_i, and p_ji(r⁰) denotes the probability of forwarding a packet to ST A_i from ST Aj ∈N₁ⁱ(r⁰).

The arrival rate Λi is theeffective arrival rate atST Ai that has two components -i) arrival from the clients, and ii) relayed packet arrival from the neighbors. Therefore, Λ_i can be expressed as,

Λ_i = Λ_e.e_i (5.2)

The utilization factor forST Ai with data rater, denoted as ρi(r), is given by;

ρ_i(r) = Λ_i

µi(r) (5.3)

Suppose, c²_Ai denotes the Squared Coefficient of Variance (SCV) of the inter-arrival times at ST A_i. Using the diffusion approximation method,c²_Ai can be expressed as,

c²_Ai= 1 + X

r⁰∈R





X

ST Aj∈N₁ⁱ(r⁰)

(c²_Bj−1)p²_ji.e_j.e⁻¹_i



 (5.4)

wherec²_Bj is the SCV of the service time at ST Aj.

According to diffusion approximation of G/G/1 queuing network, the mean number of packets at ST Ai, denoted as Πi, can be expressed as follows;

Π_i = ρ_i

1−ρ_i (5.5)

where,

ρ_i= exp

− 2(1−ρ_i) c²_Ai.ρi+c²_Bi

The parameters of the G/G/1 queuing network for the multi-rate mesh networks are calculated based on a set of lemmas, as stated follows.

Lemma 5.1. The probability that a packet is forwarded from the queue of ST A_i to the queue of ST Aj at data rate r, denoted by pij(r), is given by

p_ij(r) =







2(1−_κ¹

r) Q^~

u=1

(1−Γ(u, r)) ; ST A_j ∈N₁ⁱ(r)

0; Otherwise

(5.6)

where,

Γ(u, r) = ~−(u−1)

~(κ_ru−1 −κ_ru)

Proof. Suppose, S and R are the sender and receiver mesh STAs respectively. For a successful transmission, S and R have to first reserve channel in the DTIM interval, according to the MCCA reservation procedure. S can successfully send a MCCAOP reservation request, if no other station in the transmission range ofS reserves channel that overlaps with the reservation of S. At the same time R can forward back a MCCAOP

r r r

S R

r

w w−1 1

Figure 5.2: Transmission from mesh STAS to mesh STAR

reply if no other mesh STAs transmits simultaneously, that interfere atR. Suppose,P[.]

denotes the probability of an event occurred. Then,

P[Packet is forwarded successfully at rater] =P[No overlapping channel reservation atS]×

P[No other mesh STAs interfere with R]

(5.7) Overlapping channel reservation may occur either due to a transmission, or a reception, in the transmission range ofS. Therefore,

P[No overlapping channel reservation at S] = 2(1− 1

κ_r) (5.8)

To find out that no other mesh STAs transmits that cause interference at R, let us consider Fig. 5.2. The dotted regions show the area where a particular data rate can sustain. Suppose, <(r) denotes the region where data rate r can sustain. As discussed earlier, transmission range increases as the data rate decreases. Therefore

<(r_~)⊂ <(r_~−1) ⊂...⊂ <(r₁). Starting from the minimum data rate r1 (with maximum transmission range), interference occurs atR, if the mesh STAs at<(r₁)−<(r₂) transmit at the data rater₁. Similarly, interference can occur atR, if the mesh STAs at<(r₃)− <(r₂) transmit at the data ratesr1 and r2. Proceedings this way, interference can occur at R, if the mesh STAs at <(r_~) transmits at one of the data ratesr₁, r₂, ..., r_~.

Let us consider the mesh STAs at <(ru−1)− <(r_u) whereru, ru−1 ∈ R. There is no interference atR, if none of the mesh STAs in<(r_u−1)− <(r_u) transmits at the data rates r_u, r_u+1, ..., r_~. This indicates that there are ~−(u−1) possible data rates in which the mesh STAs in <(r_u−1)− <(r_u) cannot transmit to avoid interference at R. Suppose, E₁ denotes the event that one of the nodes in<(r_u−1)− <(r_u) transmits. ThenP[E₁] is given

as,

P[E₁] = 1 κ_ru−1 −κ_ru

Suppose, the event E₂ denotes that one of the data rates in {r_u, ru+1, ..., r_~} is chosen.

P[E₂] is derived as,

P[E₂] = ~−(u−1)

~ As the eventsE₁ and E₂ are mutually exclusive,

P[E₁.E₂] =P[E₁]×P[E₂] = ~−(u−1)

~(κru−1 −κru)

Suppose, E_u denotes the event that no mesh STAs in <(r_u−1)− <(r_u) transmit at data rate r_u, r_u+1, ..., r_~. Then,

P[E_u] = 1−P[E₁.E₂] = 1− ~−(u−1)

~(κ_ru−1 −κ_ru) = 1−Γ(u, r)

Suppose, E denotes the event that no other mesh STAs interfere withR. Then, P[E] =

~

Y

u=1

P[E_u] =

~

Y

u=1

(1−Γ(u, r)) (5.9)

Using equation (5.7), equation (5.8) and equation (5.9), the value of pij(r) can be derived.

Lemma 5.2. Suppose, there are on average ϕ number of clients under every mesh STA.

The visit ratio of ST A_i, denoted ase_i, can be expressed as,

ei = 1

ϕ

1− P

r⁰∈R

~

Q

u=1

(1−Γ(u, r⁰))

(5.10)

Proof. The packets arrive at each mesh STA according to an independent and identically distributed Poisson process. Therefore, the probability that a new packet arrives atST A_i from one of its clients equals 1/ϕ. Therefore, p0i = 1/ϕ. Substituting p0i and pji in equation (5.1),

ei = 1 ϕ+ X

r⁰∈R





X

ST Aj∈N₁ⁱ(r⁰)

1 κ_r⁰

~

Y

u=1

(1−Γ(u, r⁰)).ej





In a stable and steady state queuing network, the visit ratio for all the mesh STAs become symmetrical [209]. Therefore, e_i =e_j. Substituting the value of e_j,

ei = 1 ϕ+ei

X

r⁰∈R

~

Y

u=1

(1−Γ(u, r⁰))

!

Rearranging the values,ei can be expressed in terms of equation (5.10).

Lemma 5.3. The effective arrival rate at ST A_i, denoted as Λ_i, can be expressed as,

Λ_i = Λ_e

ϕ P

r⁰∈R

~

Q

u=1

(1−Γ(u, r⁰))

Proof. The proof can be directly derived using lemma 5.2 and equation (5.2).

Suppose, O denotes the mesh STA where the packet enters the mesh network, called the originator mesh STA, andGis the mesh gate, from where the packet leaves the mesh network. The packet is forwarded fromO toG using a set of mesh STAs as intermediate relays. The next-hop relay is selected based on the data rate. Let us assume that if an intermediate relay I selects data rate r, then it forwards the packet to the next hop relay at the maximum Euclidean distance from I, where data rate r sustains. This subsection presents a backward formulation, where the data rate is first assumed, and then its effect is analyzed over the number of hops and the interference. Though the data rate is selected based on the number of hops and the interference conditions during actual protocol design, the backward formulation is helpful for analyzing the effect of different data rate selections on the network performance. For a particular scenario, the optimal data rate can be chosen that provides best network performance according to the theoretical analysis. The effect of the selected data rate over the number of hops and the interference can be expressed using following lemmas and theorems.

Lemma 5.4. A mesh STA is said to be active if it has a packet to transmit. Suppose, Ψ_i denotes number of active interfering mesh STAs for ST Ai. Then,

E[Ψ_i] = X

r⁰∈R

κ_r⁰.ρ_i(r⁰) (5.11)

E[Ψ²_i] = X

r⁰∈R

κr⁰ρi(r⁰)(1 + (κr⁰ −1)ρi(r⁰)) + 2X

u<v

κuκvρi(ru)ρi(rv) (5.12)

Proof. Suppose, Ψ_i(r) denotes the number of active interfering mesh STAs ofST A_i, when

it transmits at data rate r. Then,

Ψi= X

r⁰∈R

Ψi(r⁰)

Ψ²_i = X

r⁰∈R

Ψi(r⁰)

!2

= X

r⁰∈R

Ψ²_i(r⁰) + 2X

u<v

Ψi(ru)Ψi(rv)

It can be noted that the random variables Ψi(ru) and Ψi(rv) are independent. Taking expectation at both sides,

E[Ψi] = X

r⁰∈R

E[Ψi(r⁰)] (5.13)

E[Ψ²_i] = X

r⁰∈R

E[Ψ²_i(r⁰)] + 2X

u<v

E[Ψ_i(r_u)]E[Ψ_i(r_v)] (5.14) ST Ai can have a packet to transmit at data rater, if its utilizationρi(r)>1, that is the arrival rate is more than the service rate. Therefore Ψ_i(r) is a binomial distribution with parameters (κr, ρi(r)). This follows,

E[Ψi(r)] =κr.ρi(r); E[Ψ²_i(r)] =κrρi(r)(1 + (κi−1)ρi(r))

Replacing the value of E[Ψi(r)] in equation (5.13) and equation (5.14), equation (5.11) and equation (5.12) directly follow.

Lemma 5.4 characterizes the interference scenario expressed as the expected numbers of active interfering mesh STAs. The next theorem characterizes the service times of the mesh routers based on the interference information.

Theorem 5.1. Suppose, Lis the packet size and Xi(r)is the service time of ST Ai, when it transmits at data rate r. Then,

E[Xi(r)] =

L r

1−^L_rΛiκr P

r⁰∈R\{r}

κ_r⁰E[Xi(r⁰)] (5.15) E[X_i²(r)] = L²

r² E[Ψ²_i] + 2E[Ψi] + 1

(5.16) Proof. The average service time of ST Ai has two components - the time when the neighbors ofST A_i reserve channel in the DTIM interval and the time whenST A_ireserves channel in the DTIM interval with data rate r. Therefore,

X_i(r) = L

rΨ_i+L r; X_i²(r) = L²

r²Ψ²_i + 2L²

r²Ψi+L² r²

Taking expectation at both sides, and using equation (5.11) yield, E[Xi(r)] = L

r X

r⁰∈R

κ_r⁰ρi(r⁰) +L r

Now, ρ_i(r) = Λ_iE[X_i(r)]. Therefore, E[X_i(r)] = L

r X

r⁰∈R

κ_r⁰Λ_iE[X_i(r⁰)] +L r

= L

rΛ_iκ_rE[X_i(r)] X

r⁰∈R\{r}

κ_r⁰E[X_i(r⁰)] + L r

Rearranging the values, equation (5.15) yields. Similarly equation (5.16) can be derived by taking expectation at both sides of the equation of X_i²(r) and using equation (5.12).

The SCV of service time of ST A_i can be calculated using equation (5.15) and equation (5.16), as follows,

c²_Bi = E[X_i²(r)]−E[Xi(r)]²

E[X_i(r)]² (5.17)

Using the value ofc²_Bi, the value ofc²_Aiand Π_ican be calculated through equation (5.4) and equation (5.5). Once the service rates of the individual mesh STAs are determined based on the interference information, the next task is to determine the average number of hops between the end-pairs. Following theorem characterizes the average number of hops required to transmit a packet, based on the selected data rates at intermediate mesh STAs.

Theorem 5.2. Suppose, D_OG be the Euclidean distance between the packet originator O and the mesh gateG. The average number of hops between O and G, denoted asS_OG, is given by,

SOG= 1

~ X

r⁰∈R

DOG

C(r⁰) (5.18)

where C(r⁰) denotes transmission range with data rater⁰.

Proof. If the packet is forwarded using data rater, then average number of hops between OandGis given byD_OG/C(r). Every intermediate mesh STA can select a data rate from R, and there are~numbers of supported data rates. Therefore, averaging over number of supported data rates, equation (5.18) yields.

Finally, the end-to-end delay is calculated based on the selected data rate, the number of hops between the end pairs (as characterize in Theorem 5.2), and the interference scenario (expressed through the service time, as presented in Theorem 5.1).

Theorem 5.3. In the steady state mesh network with multi-rate support, the average end-to-end delay from mesh STA O to mesh gate G, denoted as δ_OG, is given as,

δ_OG= ϕΠ

~Λ_e X

r⁰∈R

D_OG C(r⁰)

~

Y

u=1

(1−Γ(u, r⁰))

!

(5.19) where Π denotes average number of packets in a mesh STA at steady state. According to diffusion approximation of queuing network, at steady state, Π = Πi.

Proof. Suppose, δ_i denotes the average packet delay at ST A_i. According to Little’s law, δi= Πi/Λi. Therefore, the average end-to-end delay can be represented as,δOG=SOG×δ_i. Replacing these values using Lemma 5.3 and Theorem 5.2, equation (5.19) yields.

The steady state maximum achievable throughput of the mesh routers can also be characterized, as expressed in the following theorem.

Theorem 5.4. In the steady state mesh network with multi-rate support, the maximum achievable throughput for ST Ai, denoted asZi, is given as,

Zi = ϕrB

L(1 +κrA) (5.20)

where,

A= X

r⁰∈R\{r}

κr⁰E[Xi(r⁰)];

B = X

r⁰∈R

Y~

u=1

(1−Γ(u, r⁰))

!

Proof. The maximum achievable throughput of ST A_i is the maximum value of packet arrival rate at clients (Λe), for which average end-to-end delay remains finite. According to diffusion approximation of G/G/1 queue, in order to have finite delay, following inequality must be satisfied [209].

Λ_iE[X_i(r)]<1 Based on equation (5.15), replacing the value of E[X_i(r)],

Λ_i

L r

1− ^L_rΛ_iκ_rA <1 ⇒

L r 1

Λi − ^L_rκ_rA <1

⇒ 1 Λ_i > L

r(1 +κrA) ⇒Λ_i < r L(1 +κ_rA)

0 20 40 60 80 100 120

0 50 100 150 200 250 300 350 400

End-to-end Delay (ms)

Distance (feet) Theory, min, 6 Mbps Simulation, min, 6 Mbps Theory, max, 6 Mbps Simulation, max, 6 Mbps Simulation, Confidence Factor

Figure 5.3: Theory versus Simulation (6 Mbps)

Replacing the value of Λi using Lemma 5.3 and rearranging, Λ_e < ϕrB

L(1 +κ_rA)

This leads to equation (5.20).

For a data rate r, equation (5.19) shows the effect of the selected data rate on the end-to-end delay based on the number of hops and the interference information, according to the transmission ranges of the selected data rates. Similarly equation (5.20) shows the effect of the selected data rate on maximum achievable throughput. The rate- hop-interference trade-off has been shown in the next subsection using numerical results obtained from the theoretical analysis.