Advanced Wireless Networks - 4G Technologies

348 MOBILITY MANAGEMENT

of a mobile unit and the scheme used to construct its MLC. We then describe the algorithm used to estimate the expected times of arrival and departure of the mobile unit to a given cell within the MLC [84].

The direction-prediction method used by MLC to predict the mobile user’s direction is based on the history of its movement. It is clear, however, that the prediction method used should not be greatly affected by small deviations in the mobile direction. Furthermore, the method should converge rapidly to the new direction of the mobile unit. To take the above properties into consideration, a first-order autoregressive filter, with a smoothing factor α, is used. More specifically, let D0 be the current direction of the mobile unit when the call is made. Notice that, when the mobile is stationary within a cell, it is assumed that the current cell is the only member of the MLC, so reservations are done only within the current cell.

the prediction error, and σs is the average of the past square prediction errors at time s. σs can be expressed as σs+1 = cEs2 + (1 − c) σs .

The directional probability, at any point in time t, of any cell being visited next by a mobile unit, can be derived based on the current cell, where the mobile resides, and the

estimated direction ˜ t of the mobile unit at time . The basic property of this probability

D t

distribution is that for a given direction, the cell that lies on the estimated direction from the current cell has the highest probability of being visited in the future [83]. Consider a mobile unit currently residing at cell i coming from cell m and let j = 1, 2, . . . , represent a set of adjacent cells to cell i. Each cell j is situated at an angle ωi j from the x-axis passing by the center of cell i, as presented in Figure 11.30. If we define the directional path from

Figure 11.30 Parameters used to calculate the directional probability.

where φi j is an integer representing the deviation angle between the straight path to destination and the directional path from i to j, while θi j represents the angle between the directional path from m to i and the directional path from i to j.

a cell at the same ring as j with respect to i. A cell k is said to be at ring L with respect to cell i if it is located at a ring L cells away from i. For a given cell i, the directional probabilities Pi→ j provide the basis upon which MLCs are formed as the mobile units moves across the network.

11.4.2.1 Forming the most likely cluster

Starting from the cell where the call originated, a mobile unit is expected to progress toward its destination. The mobile unit, however, can temporarily deviate from its longterm direction to the destination, but is expected to converge back at some point in time toward its destination. This mobility behavior can be used to determine the cells that are likely to be visited by a mobile unit.

Let us define the forward span as the set of cells situated within an angle with respect

to the estimated direction ˜ t of the mobile unit as illustrated in Figure 11.31. Based on the

D

directional probabilities and the definition of a forward span, the MLC of a given mobile unit u currently located at cell i, denoted as CiMLC (u), can be expressed as CiMLC(u) = {cells j | φi j ≤ δi , j = 1, 2, . . .} where φi j is the deviation angle between the straight path to destination and the directional path from i to j. The angle δi is defined such that Pi→ j ≥ μ, where μ represents a system defined threshold on the likelihood that cell is to be visited. More specifically, δi can be expressed as δi = max |φi j | such that Pi→ j ≥ μ.

The next step in the process of forming the MLC is to decide on the size of the MLC window, WMLC, which represents the number of adjacent rings of cells to be included in the MLC. Define Ringi, j to be the ring at which cell j is located with respect to cell i. Therefore, a cell k is included in CiMLC (u), if Ringi, k ≤ WLMC which gives CiMLC(u) = {cells j | i j ≤ δi , and Ringi, j ≤ WMLC j = 1, 2, . . .}. The size of the MLC window has a strong impact

Forward span

Current cell i

˜

Dt

Figure 11.31 Definition of the MLC.

is in E if and only if j is a reachable direct neighbor of i. Each directed edge is (vi , v j ) in G is assigned a cost 1/ Pi→ j .

A path between MLC cells i and k is defined as a sequence of edges (vi , vi+1), (vi+1, vi+2), . . . , (vk−1, vk ). The cost of a path between MLC cells i and k is derived from the cost of its edges so that the least costly path represents the most likely path to be followed by the mobile. A k-shortest paths algorithm [86] is then used to obtain the set K of k-most likely paths to be followed by the mobile unit.

For each path K between MLC cell i and j, we define the path residence time as the sum of the residence time of each cell in the path. Let s and 1 in K , represent the paths with the shortest and longest path residence time, respectively. s is used to derive the expected earliest arrival time , while 1 is used to derive expected latest arrival TLA(i, j). So, TEA(i, j) and TLA(i, j) can be expressed, respectively, as

respectively. Smax(k) and Smin(k) are provided by the network support based on the observed

mobile units’ speeds. d(m, k, n) is the main distance within cell k given that cells m, k, and n are three consecutive cells in the path. The value of d(m, k, n) depends on whether cell k is the cell where the call originated, an intermediate cell or the last cell in the path, i.e. cell j

When k is the originating cell, the pdf fY (y) of the distance Y , within cell k as shown in Figure 11.32, is derived, assuming that the mobile units are evenly spread over a cell area of

Cell k’s border

Position of the

mobile unit

Figure 11.32 Distance Y in originating cell k.

352 MOBILITY MANAGEMENT

radius R travel along a constant direction within the cell and can exit from the cell from any point along the border with cell n. Therefore, the position of the mobile unit is determined by the angle ν and the distance r from the center of the cell. ν is uniformly distributed between 0 and 2π ; r is uniformly distributed between 0 and R. Since ν is uniformly distributed, φ is also uniformly distributed between 0 and π . Therefore, d (k, n) is equal to the mean distance E [Y ] of the PDF fY (y). Based on these assumptions the PDF fY (y)in a cell where the call originates can be obtained using the standard methods as described in [75]:

which gives

When k is an intermediate cell, the PDF fY (y) of the distance Y , within cell k, as shown in Figure 11.33, is derived assuming that the mobile units enter cell k from cell m at any point along the arc AB of cell k. This arc is defined by the angles β1 and β2: The mobile travels along a constant direction within the cell and can exit from cell k to n from any point along the arc CD of cell k, which is defined by the angles ω1 and ω2. The direction of the mobile is indicated by the angle φ, which is uniformly distributed; d (m, k, n) is equal to the mean distance E [Y ] of the PDF fY (y), which is derived in Aljadhai and Znati [84] (see also the

Cell n’s border

B

A

Figure 11.33 Distance Y in an intermediate cell k.

354MOBILITY MANAGEMENT

(6)A reference scheme, referred to as clairvoyant scheme (CS), is introduced. In this scheme, the exact behavior of every mobile unit is assumed to be known at the admission time. CS reserves bandwidth in exactly the cells that the mobile unit will visit and for the exact residence time interval in every cell. Therefore, CS produces the maximum utilization and minimum blocking ratio for a specific dropping ratio, which is zero in this case.

Since mobile units can appear anywhere along a cell, the residence time within the initial cell (the cell in which the call originates) is selected to be uniformly distributed between zero, and a duration equal to cell length/speed. The initial direction probability is assumed to be 0.5 for both possible directions, i.e. left and right directions. After the first handoff, the direction and position of the call become known and, therefore, it is possible to determine the arrival and departure time in other cells.

Figure 11.34 shows the blocking ratio and Figure 11.35 utilization of the three schemes as functions of the call arrival rate. As expected, CS produces the maximum utilization and minimum blocking ratio assuming a zero dropping ratio condition. The utilization in the MLC CAC is better than SC as the call arrival rate exceeds 0.06 for all mean call holding times. Moreover, the call blocking in MLC CAC is much less than that of the SC scheme. This behavior shows that, by simply reserving bandwidth between the earliest arrival time and latest departure time at a cell, the MLC scheme accepts more calls and increases utilization. Moreover, the increase in the utilization in the SC scheme is very slow when the call arrival rate is greater than 0.06. The reason for this behavior is that the SC bases its estimates on the exponential holding time PDF, which decreases as time increases. Therefore, the bandwidth estimates decreases as the distance to a future cell increases. As a result, the chance of dropping the call in subsequent cells is increased unless the minimum survivability estimate is increased. In Figures 11.34 and 11.35, the MLC CAC always outperforms the SC regardless of the mean holding time.

Figure 11.34 Blocking ratio in three systems.

Call arrival rate (call/cell/s)

Figure 11.35 Utilization in three systems.

APPENDIX: DISTANCE CALCULATION IN AN INTERMEDIATE CELL

Given an intermediate cell on the path of the mobile unit, the PDF of the distance can be derived based on the angles β and ω, as shown in Figure 11.33. The entry point to the cell is assumed to be the point E, as shown in Figure 11.33. The mobile unit move in a direction evenly distributed leading to the next cell (Figure 11.33), where 2ψ is the range of the direction angle φ. The angle β is uniformly distributed between β1 and β2. Therefore, fβ (β) is

The CDF of the above function can be represented as