Analytic Model

This section proposes an analytic model to study the LU costs for mobile telecommuni-cations networks with overlapping LAs. Suppose that an MS makes M cell movement before it leaves an LA. For each of the four policies described in section 2.2, we derive the expected number E[M]. It is clear that the larger the E [M] value, the better the performance.

2.3.1 Case 1: 0 ≤ K < ^N₂

Figure 2.4 illustrates the state transition diagram for MS cell movement in an LA, where 0 ≤ K < ^N₂. In this diagram, state Init represents that the MS moves into the LA in the

steady state. State j represents that the MS resides in cell j of the LA, where 1 ≤ j ≤ N.

Two virtual states, 0 and N + 1, are the absorbing states representing that the MS moves out of the LA from cell 1 and from cell N, respectively. For 1 ≤ j ≤ N, the MS moves from state j to state j + 1 with probability p, and the MS moves from state j to state j − 1 with probability 1 − p. As mentioned before, the entrance cell can be cell K + 1 (cell N − K) of the new LA when the MS leaves the old LA from the right-hand side (the left-hand side). Let q be the probability that the MS moves from the old LA to the new LA through the entrance cell K + 1. Then the MS moves from state Init to state K + 1 with probability q and to state N − K with probability 1 − q. Note that q is affected by the routing probability p, the overlapping degree K and the LA size N. We will derive q later.

Starting from the entrance cell j, let Nj be the number of cell movement before the MS leaves the LA. The expected number E[M] is

E[M] = qE[N_K+1] + (1 − q)E[N_{N −K}]. (2.2) We model the MS cell movement as the Gambler’s Ruin Problem [64] to solve E[Nj].

Let αj be the probability that starting from cell j, the MS will reach state N +1 before reaching state 0 (i.e., the MS moves out from the right-hand side). From Figure 2.4, we obtain the following recurrence relation for α_j

α_j = pα_j+1+ (1 − p)α_j−1 for j = 1, 2, · · · , N. (2.3) Since α0 = 0 and αN +1= 1, (2.3) is solved to yield

αj =

























1 −

Ã1 − p p

!_j

1 −

Ã1 − p p

!_{N +1} for p 6= ¹₂

j

N + 1 for p = ¹₂

. (2.4)

Define a random variable Xy as follows:

X_y =

( −1 if the MS moves left at the yth cell movement 1 if the MS moves right at the yth cell movement .

Then E[Xy] value can be a positive or a negative number, and the sign of E[Xy] indicates the direction of MS movement. By using the Wald’s Equation [64], we have

If p approaches ¹₂, E[N_j] can be derived by applying the L’Hospital’s Rule [67] to (2.7), which yields

p→lim¹₂ E[Nj] = (N + 1)j − j². (2.8) To derive q, Figure 2.5 modifies the state diagram in Figure 2.4 by removing the absorbing states 0 and N +1, and adding the transitions from state N to state K+1 (with probability p) and from state 1 to state N − K (with probability 1 − p). When the MS moves out of the current LA from the right-hand side (the left-hand side), the process moves from state N to state K + 1 (from state 1 to state N − K). In other words, the MS moves from cell N (cell 1) of the old LA to cell K + 1 (cell N − K) of the new LA. In this case, the MS

Figure 2.5: Modified State Transition Diagram for K-degree Overlapping LA Configura-tion (the LA Size is N, and 0 ≤ K < ^N₂)

would leave the current LA from the right-hand side boundary and move to cell K + 1 of new LA with probability qα_K+1(i.e., the probability that the MS moves into entrance cell K + 1 and then moves out the LA from the right-hand side) plus (1 − q)α_{N −K} (i.e., the probability that the MS moves into entrance cell N − K and then moves out the LA from the right-hand side). Since the MS moves from state Init to state K + 1 with probability q, we have the following equation

q = qα_K+1+ (1 − q)α_{N −K}, or equivalently

q = α_{N −K} 1 − αK+1+ αN −K

. (2.9)

Substituting (2.4) and (2.7) - (2.9) into (2.2), E[M] is expressed as

E[M] =























à α_{N −K} 1 − α_K+1+ α_{N −K}

! "

(N + 1)α_K+1− K − 1 2p − 1

#

+

à 1 − αK+1

1 − α_K+1+ α_{N −K}

! "

(N + 1)αN −K− N + K 2p − 1

#

for p 6= ¹₂

(N − K) × (K + 1) for p = ¹₂

,

(2.10)

where

In this case, an entrance cell is covered by two or more LAs. Therefore, after the MS leaves the old LA, an LA selection policy is required to select the new LA. The E [M]

values for the four policies are derived as follows.

• MaxOL: The MS always chooses a new LA with maximum overlapping with the old LA. When the MS moves out of LA i from the right-hand side (the left-hand side), it registers to LA i + 1 (LA i − 1). Clearly, the LA selected in this policy is the same as that selected in case 0 ≤ K < ^N₂ described in Section 2.3.1. Therefore, the expected number of MS cell movement in an LA is expressed in (2.10).

• Central: After moving out of LA i, the MS always registers to the LA whose central cell is closest to the entrance cell. The selected LA is the ^lN₂^Om-th LA away from LA i. That is, the entrance cell is cell N + 1 −^lN₂^Om(N − K) of LA i +^lN₂^Om in the right-hand side or cell^lN₂^Om(N − K) of LA i −^lN₂^Omin the left-hand side. The state transition diagram for the Central policy is shown in Figure 2.6, where the MS moves from state Init to state N +1−^lN₂^Om(N −K) with probability q⁰ and to state

lNO

2

m(N − K) with probability 1 − q⁰. Following the derivation in Section 2.3.1, we have

Figure 2.6: State Transition Diagram for the Central Policy (^N₂ ≤ K < N )

NO

−p 1

NO

−p 1

NO

−p 1

NO

p

NO

p

NO

p

Figure 2.7: State Transition Diagram for the Random Policy (^N₂ ≤ K < N )

• Random: The MS randomly registers to one LA covering the entrance cell. Let ql

(qr) be the probability that the MS moves from a left-hand side LA (right-hand side LA) to the new LA through the entrance cell. The state transition diagram for the Random policy is shown in Figure 2.7, where the MS moves from state Init to state N + 1 − m(N − K) with probability q_l and to state m(N − K) with probability q_r, where 1 ≤ m ≤ N_O. From Figure 2.7, E[M] is expressed as

E[M] = q_l





NO

X

m=1

E^hNN +1−m(N −K)

i



+ q_r





NO

X

m=1

E^hN_{m(N −K)}ⁱ



 . (2.12) Since the entrance cell is covered by N_O LAs, when the MS leaves the old LA, the process moves from state N (state 1) to state N + 1 − m(N − K) (state m(N − K)) with probability _N^p

O (^1−p_N

O), where 1 ≤ m ≤ N_O; see Figure 2.7. The balance

equations for probabilities q_l and q_r are

• MinOL: After leaving LA i, the MS chooses the farthest LA of the entrance cell from LA i. The entrance cell is cell N + 1 − NO(N − K) of LA i + NO in the right-hand side or cell N_O(N − K) of LA i − N_O in the left-hand side. The state transition diagram for the MinOL policy is shown in Figure 2.8. In this figure,

Figure 2.8: State Transition Diagram for the MinOL Policy (^N₂ ≤ K < N ) (see Table 2.1). The errors between the analytic and simulation models are under 1%.