Fractional Hybrid Movement-distance-based Location Update with Selective Paging

In this paper we propose a fractional movement-distance-based location update scheme for Personal Communication Service PCS network. Indeed it is a location update scheme which is mobile terminal (MT ) dependent. Each MT stores a set of cells within a distance H , in terms of cells, from the cell where the last update occurred. The MT can then move freely within a ring distance Hl = H − 1 without the need for updating (the movement counter is reset —disabled—). When the MT moves within a ring distance between H + 1 and D − 1 the movement counter is enabled, i.e., each time the MT visits a new cell within the mentioned distance, the movement counter of the MT is increased in one unit. When the MT enters into cell ring H , the counter is increased with probability p, or it is reset with probability r = 1− p. The state of the counter, enabled or disabled, is maintained while the MT is moving in cell ring H . Each time the counter of the MT reaches the threshold d (or d + 1, to be specified later on), a location update is triggered and a new set of cells is memorized. In a similar way to the fractional movement-based scheme, [7], in our proposed scheme, the movement threshold is a real number between d and d + 1. It is shown that the optimal value for the threshold counter of the MT is a real number, not necessarily integer, d + p. That optimal value minimizes the total cost of location updates and paging per call arrival. This minimum is specially significative when selective paging is considered. Selective paging decreases the location tracking cost under a small increase in the allowable paging delay. Indeed, our proposal provides a unified view of the movement-based and the distance-based schemes. In the movement-based scheme, each mobile terminal only keeps a counter of the number of cells visited, i.e., the set of cells (records) the MT keeps in its local memory is an empty set, and the threshold counter is d = D. In the distance-based scheme no movement counter is required but the mobile terminal keeps in its local memory the identification of all cells which are within a distance D − 1. Our analytical model provides an easily programmable tool to evaluate the mentioned trade-off. 1. Scenario, system description and formulation The PCN coverage area is partitioned into hexagonal cells of the same size. The cell residence time of a MT follows the Gamma distribution, with mean value 1/λm and variance Vr. In [6] it is shown that the cell residence time can be described by the generalized Gamma distribution. The Laplace transform of Gamma distribution, f∗ m(s) = ∫∞ 0 fm(t)e−stdt, is given by f∗ m(s) = ( λmγ s + λmγ )γ ; γ = 1 Vrλm (1) Here we will use a random walk mobility model. When the MT leaves a cell, the probability to visit a new cell is proportional to the common perimeter with this new cell. The number of records (cell identities) in the memory of the MT , M , is a non empty set of cells composed by H + 1 consecutive cell rings, from the center cell (cell ring 0) up to cell ring H . See the hexagonal cell layout, figure 1, where configurations (H,D) = (4, 8) is depicted and M = 3H + 3H + 1. Although a general incoming call process to the MT (inter-arrival time distribution) could be considered, [8], for the sake of simplicity here we have assumed Poisson process with parameter λc. The call duration is negligible compared with the interarrival time duration, such that the busy line effect does not occur [3] (i.e., there is no new phone call to a mobile terminal when it is in conversation). Under this assumption, the probability that there are z boundary crossing between two call arrivals, α(z, a, θ), it is given by, [5]: α(z, a, θ) = { 1− 1 θ (1− a); z = 0 1 θ (1− a)2az−1; z > 0 (2) where a = f∗ m(λc), and θ = λc/λm is the call-to-mobility ratio, CMR, defined in [2]. 2. Location update cost Figure 1 represents the Markov chain for a generic (symmetric) cell layout configuration, mesh or hexagonal random walk. Each state represents a cell ring, except for cell ring H which is split into two states Hr and Hf . Cleary Hl + 1 = Hr and Hf = Hu − 1. We say that the MT is in state Si if it is roaming in a cell ring i. The set of cells that the MT keeps in its memory is from cell ring0 cell up to cell ring H cell. Starting from a state S0, the MT counter is set to zero. The counter is not increased while the Markov chain makes transitions between the set of states S0, S1, .., SHl , SHr . The MT counter is set to one when either, one of the transitions, SHl → SHf or SHr → SHu occur. In the first (second) case, i.e., after the transition SHl → SHf (SHr → SHu ), if an absorption into states SHl or SHr (SHl or SHr ) occurs before d = D −Hl (d − 1 = D − Hr) consecutive movements, the MT resets its movement counter; and when the counter reaches the movement threshold d + 1 (d) without being absorbed into states SHl or SHr (SHl or SHr ), the MT triggers a LU message. For a given number of movements, z, we are interested in the number of LU messages triggered by the MT . To that purpose, we will use first passage, first return and taboo probabilities. Following the terminolgy in [1] we call f (n) i,j the conditional probability that state Sj is avoided at times 1, 2, .., n− 1 and entered at time n, given that state Si is occupied initially. Similarly {G}f (n) i,j is the taboo probability that the Markov chain enters state Sj for the first time at the nth step, having initially started from state Si and avoiding the set of states G at times 1, 2, .., n − 1. Let Pnab,l(Hl, D) (Pnab,r(Hr, D)) denote the probability that after visiting state SHl (SHr ) the MT triggers a LU within the next consecutive d = D − H + 1 (d − 1 = D − H) movements. Pnab,l(Hl, D) and Pnab,r(Hr, D) are given by Pnab,l(Hl, D) = 1− f (1) Hl,Hl−1 − D−Hl ∑ n=1 [ {Hl−1,Hr}f (n) Hl,Hl + {Hl−1,Hr}f (n) Hl,Hr ] (3) Pnab,r(Hr, D) = 1− D−Hr ∑ n=1 [ {Hl,Hr}f (n) Hr,Hl + {Hl,Hr}f (n) Hr,Hr ] (4) Let Pm,k(z,D) denote the probability of having k LU messages triggered in z movements, given that state Sm is occupied initially. We can write the following recursive relationships. P0,k(z, D) =    1; k = 0, z < D ∑1 l=0 f (1) 0,l Pl,0(z − 1, D); k = 0, z ≥ D 0; k > 0, z < kD ∑1 l=0 f (1) 0,l Pl,k(z − 1, D); k > 0, z ≥ kD (5) Pm,k(z, D) =    1; k = 0, z < D −m ∑m+1 l=m−1 f (1) m,lPl,0(z − 1, D); k = 0, z ≥ D −m 0; k > 0, z < kD −m ∑m+1 l=m−1 f (1) m,lPl,k(z − 1, D); k > 0, z ≥ kD −m m = 1, 2, .., Hl − 1 (6) PHl,k(z, D) =    1; k = 0, z < D −Hl ∑D−Hl n=1 [ {Hl−1,Hr}f (n) Hl,Hl PHl,0(z − n,D) + {Hl−1,Hr}f (n) Hl,Hr PHr,0(z − n,D) ] +f (1) Hl,Hl−1PHl−1,k(z − 1, D); k = 0, z ≥ D −Hl 0; k > 0, z < kD −Hl ∑D−Hl n=1 [ {Hl−1,Hr}f (n) Hl,Hl PHl,k(z − n,D) + {Hl−1,Hr}f (n) Hl,Hr PHr,k(z − n,D) ] +f (1) Hl,Hl−1PHl−1,k(z − 1, D) + Pnab,l(Hl, D)P0,k−1(z − d, D); k > 0, z ≥ kD −Hl (7) PHr,k(z,D) =    1; k = 0, z < D −Hr ∑D−Hr n=1 [ {Hl,Hr}f (n) Hr,Hl PHl,0(z − n,D) + {Hl,Hr}f (n) Hr,Hr PHr,0(z − n,D) ] ; k = 0, z ≥ D −Hr 0; k > 0, z < kD −Hr ∑D−Hr n=1 [ {Hl,Hr}f (n) Hr,Hl PHl,k(z − n,D) + {Hl,Hr}f (n) Hr,Hr PHr,k(z − n,D) ] +Pnab,r(Hr, D)P0,k−1(z − d,D); k > 0, z ≥ kD −Hr (8) Having in mind expressions (3)–(8), the location update cost can be formulated in terms of Mm(z, D) = ∑ k kPm,k(z, D). Mm(z, D) is the expected number of location update messages triggered by the MT in z movements, given that state Sm is occupied initially. For a Poisson call arrival process with rate λc the location update cost, Cu can be written as: Cu(a, θ, Hr, r,D, U) = U ∞ ∑ z=D α(z, a, θ)M0(z, D) (9) 3. Terminal paging cost For paging procedure, we use a shortest-distance-first (SDF ) partitioning scheme, [4]. Let πp,i(z,D), i ∈ [0, D − 1] denote the probability that starting at cell ring p the MT is located at cell ring i after z movements. Then, we can write the following recursive relationship π0,i(z, D) = { p (z) 0,i ; z < D ∑1 l=0 f (1) 0,l πl,i(z − 1, D); z ≥ D (10) πm,i(z, D) = { p (z) m,i; z < D −m ∑m+1 l=m−1 f (1) m,lπl,i(z − 1, D); z ≥ D −m m = 1, 2, .., Hl − 1. (11) πHl,i(z,D) =    p (z) Hl,i ; z < D −Hl ∑D−Hl n=1 [ {Hl−1,Hr}f (n) Hl,Hl πHl,i(z − n,D) + {Hl−1,Hr}f (n) Hl,Hr πHr,i(z − n, D) ] +f (1) Hl,Hl−1πHl−1,i(z − 1, D) + Pnab,l(Hl, D)π0,i(z − d,D); z ≥ D −Hl (12) πHr,i(z, D) =    p (z) Hr,i ; z < D −Hr ∑D−Hr n=1 [ {Hl,Hr}f (n) Hr,Hl πHl,i(z − n, D) + {Hl,Hr}f (n) Hr,Hr πHr,i(z − n,D) ] +Pnab,r(Hr, D)π0,i(z − d, D); z ≥ D −Hr (13) From (10)–(13), we can get the probability that the MT is in cell ring i when a call arrival occurs, q0,i(a,D) = ∑∞ z=0 α(z, a, θ)π0,i(z, D). After that we can compute expressions (11), (12), (13) and (14) of [4]. So the expected terminal paging cost per call arrival, denoted by Cv , is Cv(a, θ, Hr, r,D, V ) = V l−1 ∑ k=0 [ ∑ ri∈Ak q0,i(a,D) k ∑ m=0 ∑ rj∈Am g(j) ] (14) In (14), l = min (η, D), η is the maximum allowable paging delay, g(i) is given by expression (1) of [4], and subarea Aj contains rings sj to ej where sj and ej are the indices of the first and the last rings in subarea Aj . ej and sj are also given in [4]. 4. Performance evaluation and numerical results Clearly, the total cost per call arrival is defined as CT (a, θ, Hr, r,D, U, V ) = Cu(a, θ, Hr, r,D, U) + Cv(a, θ,Hr, r,D, V ) (15) This cost is evaluated for several set of parameters. The parameters include the update cost U , the polling cost V , the location update distance thresholds, D, the location update movement threshold, between D−Hr and D−Hl, the maximum paging delay η, the call-to-mobility ratio θ = CMR, and the variance, V r, of the cell residence time. Numerical results will be provided in the final version of the paper. Minimum cost is achieved when the location update movement threshold, or equivalently, when the parameter r is a real number such that 0 < r < 1.