5.3 Analytical Model
5.3.3 Markov-model
States
One specific mobile node in a mobility network that uses Hierarchical Paging can be mod-eled using a discrete-time Markov-chain. The Markov-chain has four states: P0, P1, P2
and P3. The meaning of the states is the following (for reference see Section 5.1.2):
• P0: The exact location of the mobile node is currently known; at the top-level it is known which subnetwork the mobile node is currently staying in, and at the bottom level it is known which cell the mobile node is currently staying in.
• P1: The top-level mobility management has subnetwork-level location information;
at the top-level it is known which subnetwork the mobile node is currently staying in, but it is not known which cell the mobile node is in.
• P2: The subnetwork which the mobile node is staying in is not known at the top level, but (at the bottom level) the root node of the subnetwork knows the cell which the mobile node is staying in.
• P3: The only information the network has is that the root node of the subnetwork that the mobile node is staying in knows that the mobile is in that specific subnetwork;
the subnetwork which the mobile node is staying in is not known at the top level, and the cell that the mobile node is staying in is not known at the bottom level.
Whenever there is a handover or an incoming call, there is a transition in the Markov-chain. Let p denote the probability that the next event is a handover and not a received call. According to the considerations in Subsection 5.3.2,
p= λ
λ+µ, (5.1)
where p is a parameter of our model, we have no way of affecting it by repartitioning the subnetwork or by changing the mobility management in any other way. This parameter is sometimes referred to as “mobility ratio”[8].
Transition Probabilities
Let’s start fromP0; the exact position of the mobile is known. While there are no handovers just incoming calls, the position remains known, thus the states remains P0. If there is a handover (with probability p), it is an inter-subnetwork handover (going to P2) with probability p
(k/L), and is an intra-subnetwork handover (to state P1) with probability 1−p
(k/L). Figure 5.1 shows the probabilities of all the possible transitions. Generally, we move to P2 with a probability of p(p
(k/L)), which means that no matter what kind of position information the network had before; after an inter-subnetwork handover the top-level will have no exact position information, and the bottom-level will (the mobile node sends the location update message from a cell, and the bottom-level will know this cell, so it will have exact location information until the mobile node moves away.
Costs
The costs of the various mechanisms are defined in this section. These costs are of course not absolute costs, but relative ones that are used for comparison of different solutions or the same solution with different parameter values.
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Figure 5.1: Hierarchical paging Markov-model transition probabilities
In this hierarchical paging scheme an intra-subnetwork handover has a cost of zero. No location update messages are sent, no databases are updated, no entity is notified.
In case of an inter-subnetwork handover, the root nodes of the old and new subnetworks have to be notified about the handover, thus a constant cost of 2 will be assigned to this process.
The cost of receiving an incoming call or packet is made up of the cost of determining the exact location of the mobile node and the cost of receiving the call itself. We have no way of affecting the cost of receiving the call, and it is always there, so we will only consider the cost of determining the exact location of the mobile node.
The cost of determining the location depends on what kind of location information the network had when the call (or packet) arrived. Note, that when the exact position
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Figure 5.2: Hierarchical paging Markov-model transition probabilities and costs is determined, these are transitions to P0. If we have exact location information an all levels, then the cost is zero (loop from P0 to P0). When the subnetwork is known, but not the exact cell, a subnetwork-wide paging has to be carried out (from P1 to P0) which involves L/k cells, thus a cost of L/k is assigned. When the cell within the subnetwork is known, but the subnetwork is not (form P2) a top-level paging takes place, k subnetworks are involved, thus a cost of k is assigned. If neither the subnetwork nor the exact cell within the subnetwork is known, then both top-level and bottom-level paging operations are carried out, thus the cost is k+L/k.
Figure 5.2 shows all the probabilities and non-zero costs of both handovers and the determination of exact locations. The cost is always on the right of the “pipe” symbol.
Stable State
As the Markov-chain is finite, aperiodic and irreducible (as 0 < p < 1), it is stable. The stable state can be computed easily:
P0 = 1−p,
P1 =−−p+ q
k
Lp+p2− q
k Lp2 1−p+
q
k Lp
,
P2 = rk
Lp, P3 =−(−1 +
qk L)
qk Lp2 1−p+
qk Lp
,
where P0...P3 denotes the probability of being in stateP0...P3 in the stable state of the Markov-chain.
Average Cost of an Event
Events in our model are handovers and incoming calls. As explained in Section 5.3.3, we have no way to affect what events occur and when, it is an input parameter. So, the average cost of an event describes how efficient our solution is. The cost of an average event can be determined using the probabilities of the stable state and the costs of transitions. It is a rather complicated, but closed form solution:
Cost(k, L, p) = (5.2) k
rk
L(1−p)p− µ
−1 + q
k L
¶q
k L
¡k+Lk¢
(1−p)p2 1−p+
q
k Lp
− 2k
µ
−1 + q
k L
¶ p3 L
µ
1−p+ q
k Lp
¶ −
L(1−p) µ
−p+ q
k
Lp+p2 − q
k Lp2
¶
k µ
1−p+ q
k Lp
¶
− 2
qk Lp
µ
−p+ qk
Lp+p2− qk
Lp2
¶
1−p+ q
k Lp
.
The smaller the cost is, the more efficient our solution is (considering all other para-meters are constant).
Cost Coefficients
The cost of an update compared to a search procedure can be weighted by introducing various cost coefficients or factors. When calculating the total cost, the update costs are weighted (multiplied) by the update cost coefficient and the search costs (paging) are weighted by the search cost coefficient. These factors of course change the formula of the average cost of an event.