Part II Summary and Further Reading
8.2 Markov Chain Model for a Single Node
As always, buffer occupancies at all Markov points should add up to one:
Lbri k=0 qk+ Lbri−B k=0 hk =1 (8.22)
By solving the system (8.21) together with the Equation (8.22), we obtain the probability distribution of the bridge buffer occupancy at relevant Markov points.
The blocking probability at the bridge buffer is
PbriB = qLbri Lbri i=0 qi (8.23)
which critically affects the performance of the source cluster under acknowledged transfer, since blocked packets have to be retransmitted until acknowledged.
8.2 Markov Chain Model for a Single Node
Let the PGF of the data packet length beGp(z)=zk, where the packet size is equal tok
backoff periods. In case of non-acknowledged transfer, the PGF for the total transmission time of a data packet is Dd(z)=z2Gp(z), where the term z2 models the two CCAs. In
case of acknowledged transfer, we need to include the time to receive the acknowledgment: the time interval between the packet transmission and subsequent acknowledgment and the acknowledgment packet itself, the PGFs of which are tack(z)=z2 and Ga(z)=z,
respectively, as explained in Section 3.3. Then, the PGF for the total transmission time of the data packet under acknowledged transfer will be Dd(z)=z2Gp(z)tack(z)Ga(z),
while its mean value isDd =2+Gp(1)+tack (1)+Ga(1). The PGF of the duration of
the beacon frame is denoted as Bea(z)=z2, which assumes that the beacon contains no
payload and no GTS announcements.
The discrete-time Markov chain that models the behavior of ordinary nodes in both clusters, as well as the behavior of the bridge node, is shown in Figure 8.5 and 8.6. Although this chain is virtually identical to the one presented in Section 3.3 and shown in Figure 3.4 and 3.2, we repeat it here for convenience; of course, the values of various
0,2,W0−1 0,2,W0−2 0,2,1 0,2,0 1 0,1,0 0,0,0 1 β idle ‘delay line’ 0 Pd 0,2,W0−1 1 0,2,W0−2 0,2,1 1 0,2,0 0,1,0 β 1-β 1-β (1-Pd)α (1-Pd)(1-α) 1/8 7(1-α)/8 7α/8 1-φa-φi(no arrivals)
φi (during inactive part
of the superframe)
φa (during active part
of the superframe) θ0(success,no more packets) 1-θ0 1,2,W1-1 1,2,W1-2 1,2,1 1,2,0 m,2,Wm−1 m,2,Wm−2 m,2,1 m,2,0 m+1,0,0 1,1,0 1,0,0 m,1,0 m,0,0 1 1 1 1 β β 1-β 1-β (1-Pd)α ‘d elay line‘ m Pd 1 (1-Pd)(1-α) (1-Pd)α (1-Pd)(1-α) ‘d elay line’ 1 Pd 1 uniformly distributed among the W1 states
uniformly distributed among the Wm states
uniformly distributed among the W0 states
uniformly distributed among the W0 states
Tr Tr Tr Tr 1-θ0 1-θ0 Tr Tr θ0 θ0
Figure 8.5 Markov chain model that applies to an ordinary node in both source and sink clusters, as well as to the bridge. Adapted from J. Miˇsi´c and C. J. Fung, ‘The impact of master-slave bridge access mode on the performance of multi-cluster 802.15.4 network,’ Computer Networks51: 2411–2449,2007 Elsevier B.V.
160 CLUSTER INTERCONNECTION WITH MASTER-SLAVE BRIDGES
‘ ’
Figure 8.6 Delay lines for Figure 8.5. Adapted from J. Miˇsi´c and C. J. Fung, ‘The impact of master-slave bridge access mode on the performance of multi-cluster 802.15.4 network,’ Computer Networks51: 2411–2449,2007 Elsevier B.V.
transition probabilities will differ. We assume that this Markov chain has a stationary distribution. The state of the node at backoff unit boundaries is defined by the process
{i, c, k, d}, where
• i∈(0. . m)is the index of current backoff attempt (mis a constant defined by MAC
with default value 4).
• c∈(0,1,2)is the index of the current Clear Channel Assessment (CCA) phase. • k∈(0. . Wi−1)is the value of backoff counter, withWi being the size of backoff
window ini-th backoff attempt. The minimum window size isW0=2macMinBE, while
other values are equal toWi =W02min(i,5−macMinBE)(by default,macMinBE=3). • d ∈(0. . Dd−1) denotes the index of the state within the delay line mentioned
above; in order to reduce notational complexity, it will be shown only within the delay line and omitted elsewhere.
The Markov chain from Figure 3.4, is general in the sense that the probabilities of leaving the transmission state θ0 and leaving the idle state φa, φi are generally labeled;
in order to model the bridge, source node or sink node under acknowledged or non- acknowledged transfer, they need to take actual values. θa andθi denote the probability
that a new packet will arrive to the node which is currently in the idle state (i.e., it has no packets in its buffer) during the active and inactive part of the superframe, respectively.
θ0 is the probability that the node buffer is empty after successful packet transmission.
For clarity, we willfirst solve the general model and then substitute the values specific to bridge, source, and sink nodes, respectively.
We note that the part of the Markov chain that models thefirst backoff phase actually has two components. The sub-chain connected to the idle state with the probability φi
represents the case in which a new packet arrives to the empty buffer during the inactive portion of the superframe. In that case, thefirst backoff countdown will start immediately after the beacon and the value of the backoff counter will be in the range 0. . W0−1. The
corresponding states in the sub-chain will be denoted as xi,c,ks . On the other hand, if the
packet arrives to a node during the active portion of the superframe, the backoff countdown will start at a random position within the superframe; those states will be denoted asxi,c,k.
As these two cases will have different effect on the behavior of the medium, they have to be modeled separately.
From the Markov chain, we define the probability to access the medium as
τ = m
i=0 xi,0,0.
Then, the probability that the node will switch into the idle state isτ θ0. After setting the
balance equation for the idle state, the probability of being in the idle state is obtained as Prob(idle)=Pz=
τ θ0 φa+φi
.
If we further consider the output from the idle state, and set the balance equations for the first backoff phase after the idle state started during the inactive part of the superframe, we obtain
xs0,2,W0−k=Pz kφi
W0,1≤k≤W0 (8.24)
Thefirst backoff phase in the active part of the superframe may begin at the following moments: upon the arrival of a packet during the idle state, upon a packet transmission (but regardless of the transmission success), or after the last unsuccessful backoff phase. Let us denote the state after the last unsuccessful backoff phase withxm+1,0,0 (this is done for
clarity only, and has no physical meaning whatsoever). The state probabilities of thefirst backoff phase started in the active superframe part are represented asx0,2,k, 0≤k < W0.
The input probability for that set of states is
Ua =Pzφa+τ (1−θ0)+xm+1,0,0 =τ 1−θ0 φi φi+φa +xm+1,0,0 (8.25)
By setting the balance equations we obtain
x0,2,W0−k =k Ua
W0,1≤k≤W0 (8.26)
A similar approach can be applied to derivexi,2,k for backoff attemptsi=2,3, . . . m.
Using the transition probabilities indicated in Figure 8.5 and 8.6, we can derive the relationships between the state probabilities and solve the Markov chain. For brevity, we will omit indexd whenever it is zero, and introduce the auxiliary variables C1,C1s, C2,
162 CLUSTER INTERCONNECTION WITH MASTER-SLAVE BRIDGES
C2s,C3 andC3s through the following equations: x0,1,0 =x0,2,0(1−Pd)α+x0s,2,0 7α 8 =x0,2,0C1+x0s,2,0C1s x1,2,0 =x0,2,0(1−Pd)(1−αβ)+x0s,2,0 7 8(1−αβ)=x0,2,0C2+x0s,2,0C2s =τ θ0 φi φi+φa C2s+ 1−θ0 φi φi+φa C2 +C2xm+1,0,0 x0,0,0 =x0,2,0((1−Pd)αβ+Pd)+x0s,2,0 7 αβ 8 + 1 8 =x0,2,0C3+x0s,2,0C3s (8.27)
whereαandβ denote the success probability for thefirst and second CCA, respectively.
From the expressions that describe the Markov chain we obtain
xi,1,0 =C1C2(i−1)x1,2,0, i=1. . m xi,2,0 =C2(i−1)x1,2,0, i=1. . m xi,0,0 =C3C2(i−1)x1,2,0, i=1. . m+1 Dd−1 d=0 xi,2,0,d=xi,2,0C2i(Dd−1)/2 (8.28)
Of course, the sum of all probabilities in the Markov chain must be equal to one:
U+ W0−1 k=0 xs0,2,k+xs0,1,0+ m i=0 Wi−1 k=0 xi,2,k+ m i=0 xi,0,0(Dd−2)+ + m i=0 xi,1,0+xm+1,0,0+ m i=0 Dd−1 d=0 xi,2,0,d =1 (8.29)
which has to be solved forτ, the probability of successful transmission, averaged over the
duration of the active portion of the superframe. However, no access is possible in thefirst two, as well as in the lastDd−1 backoff periods, and the access probability has to be
scaled accordingly.
By the same token, packets that arrive to an idle node during the inactive portion of the superframe can only access the medium (i.e., be transmitted) between the third backoff period after the beacon and the W0+2-th backoff period of the superframe. (The reader
may recall thatW0=2macMinBE denotes the minimum window size for the random backoff
countdown.) From the solution of the Markov chain,x0s,2,0 is the probability that medium
access is possible in any backoff period; again, this value needs to be scaled to the time interval in which access can occur. Since the initial value for the countdown is chosen at random between 0 and W0−1, the probability of access in the third backoff period after
the beacon is τ3,1=xs0,2,0 1 W0 SM−2 W0 , (8.30)
where SM=SD−Dd +1 is the duration of the part of the superframe wherein either
interval in which packet transmission can occur. The probability that the access will occur in some other (fourth toW0+2-th) backoff period after the beacon is
τ3,2=xs0,2,0 W0−1
W0 αβ
SM−2
W0−1 (8.31)
The reason for separatingτ3,1 from τ3,2 is that the former overlaps with the transmissions
deferred from the previous superframe due to insufficient time. The probability to access the medium in this case isSM−2 times the value averaged over the entire superframe, as it can happen only in the third backoff period after the beacon. Therefore, the success probability is τ1 =(SM−2)Pd C3 τ −x0,2,0Cs3 τ2 = 1−Pd C3 (τ−x0,2,0C3s) (8.32) for deferred and non-deferred packets, respectively.
8.2.1 Case 1: ordinary node in the source cluster
Let us now discuss the performance of specific nodes in the network, starting with ordinary nodes in the source cluster.
Non-acknowledged transfer. The idle state of the Markov chain is reached when the buffer is empty after transmission, regardless of whether the packet suffered the collision or blocking by the bridge. In this case,θ0,src=π0,src, which was derived in Equation (8.14)
above. Since the packet arrival rateλis small, the probability of no arrivals during a single
backoff period can be approximated with thefirst term of the corresponding Taylor series,
exp(−λ)≈1−λ. Then, the probability of non-zero packet arrivals (in which case the node
will leave the idle state) isλwhich further givesφi,src=Psyncλandφa,src=(1−Psync)λ,
wherePsync=1−2SO−BOis the conditional probability that a packet has arrived to an idle
node during the inactive portion of the superframe.
Acknowledged transfer. In this case, the idle state is reached only if the node buffer is empty after the transmission, the transmission was successful, and the packet was accepted by the bridge. As the bridge has afinite buffer, a packet sent by an ordinary node can be rejected even though the transmission itself was successful. Therefore,
θ0,src = γsrcδsrc(1−PbriB ) Pa,src π0,src, where Pa,src= a−1 i=0 (1−γsrcδsrc(1−PbriB ))iγsrcδsrc(1−PbriB ) = 1−(1−γsrcδsrc(1−PbriB ))a
denotes the probability that the transfer is completed withina attempts. Furthermore, the
probabilities of leaving the idle state are φi,src=Psyncλand φa,src =(1−Psync)λ (since
164 CLUSTER INTERCONNECTION WITH MASTER-SLAVE BRIDGES
8.2.2 Case 2: bridge in the CSMA-CA mode
Non-acknowledged transfer. The probability to enter the idle state of the Markov chain is θ0,bri= h0 Lbri−B i=0 hi
Packets can arrive to the bridge only when it is present in the source cluster, and the packet arrival rate during this period is
λbri =nλ(1−PsrcB )γsrcδsrc.
Therefore, the probabilities to leave the idle state areφa,bri =0 andφi,bri=λbri.
Acknowledged transfer. In this case, the probability to enter the idle state of the Markov chain is θ0,bri = h0 Lbri−B i=0 hi ·γbriδbri Pa,bri
whereδbri denotes the probability that a packet transmitted by the bridge is not corrupted
by noise, and
Pa,bri = a−1 i=0
(1−γbriδbri)iγbriδbri
is the probability that transfer is completed withina attempts. Packet arrival to the bridge
is possible only when the bridge is present in the source cluster, in which case the packet arrival rate to the bridge queue is
λbri=n
λ(1−PB src)Pa,bri
1−PbriB .
Finally, the probabilities to leave the idle state areφi,bri=λbri andφa,bri =0.
8.2.3 Case 3: ordinary node in the sink cluster
Under non-acknowledged transfer, the idle state is reached when the node buffer becomes empty regardless of the success of the transmission. In that case, the probabilities to enter the idle state areθ0,snk =π0,snk,φi,snk=Psyncλ, andφa,snk =(1−Psync)λ.
Under acknowledged transfer, an ordinary node in the sink cluster can reach the idle state of the chain only if its buffer remains empty after a successful transmission, which means that
θ0,snk =
γsnkδsnk Pa,snk π0,snk
andδsnk=δbri. The idle state is left upon the arrival of a new packet, which happens with
the probability ofφa,snk=(1−Psync)λandφi,snk=Psyncλin the active and inactive state,
Now, the bridge may operate in either CSMA-CA or GTS mode. While ordinary nodes in the sink cluster use CSMA-CA medium access regardless of the bridge access mode, and the Markov chain for both modes looks virtually the same, there are two subtle differences that need to be accounted for:
• When the bridge operates in the CSMA-CA mode, ordinary nodes in the sink cluster must compete with the bridge to access the medium. As the bridge delivers the traffic from the entire source cluster (of which it is the coordinator), the arrival rate from the bridge will be much higher than that of an ordinary node in the sink cluster.
• When the bridge operates in the GTS mode, its transmissions are confined to the CAP and do not interfere with those from ordinary nodes in the sink cluster. However, the active portion of the superframe in the sink cluster is shorter because the CAP is present, and contention among ordinary nodes will increase.
The exact details of these interactions will be seen in the following.