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Advanced Wireless Networks - 4G Technologies

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328

MOBILITY MANAGEMENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Homing Base

 

 

 

 

MULTICAST

 

 

 

 

Station Re-routing

 

 

 

 

CONNECTION

 

 

 

 

 

 

 

 

 

RE-ROUTING

 

 

 

 

 

 

 

 

 

 

 

 

Virtual connection

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Tree Re-routing

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hybrid Connection

 

 

 

 

PARTIAL

 

 

 

 

Re-routing

 

 

 

 

CONNECTION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RE-ROUTING

 

 

 

 

 

 

 

HANDOFF

 

 

 

 

Nearest Common

 

 

 

 

 

 

 

 

 

 

 

 

Node Re-routing

 

 

 

 

MANAGEMENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ROUTE

 

 

InterWorking Devices

 

 

 

 

 

 

AUGMENTION

 

 

Connection Extension

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FULL

 

 

 

 

 

 

 

 

 

 

 

InterWorking Devices

 

 

 

 

 

 

CONNECTION

 

 

 

 

 

 

 

 

 

Connection Re-route

 

 

 

 

 

 

RE-ROUTING

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 11.20 WATM handoff management techniques.

potential protocols to be used in 4G, can be grouped into four categories:

(1)full connection rerouting;

(2)route augmentation;

(3)partial connection rerouting; and

(4)multicast connection rerouting.

Full connection rerouting maintains the connection by establishing a completely new route for each handoff, as if it were a brand new call [46]. Route augmentation simply extends the original connection via a hop to the MT’s next location [46]. Partial connection rerouting reestablishes certain segments of the original connection, while preserving the remainder [9]. Finally, multicast connection rerouting combines the former three techniques but includes the maintenance of potential handoff connection routes to support the original connection, reducing the time spent in finding a new route for handoff [9]. More details can be found in the above references.

11.1.2.8 Mobility management for satellite networks

In 4G integrated wireless networks the LEO satellites would cover regions where building terrestrial wireless systems are economically infeasible due to rough terrain or insufficient user population. A satellite system could also interact with terrestrial wireless network to absorb the instantaneous traffic overload of the terrestrial wireless network.

LEO satellites are usually those with altitudes between 500 and 1500 km above the Earth’s surface [70–72]. This low altitude provides small end-to-end delays and low power requirements for both the satellites and the handheld ground terminals. In addition, intersatellite links (ISL) make it possible to route a connection through the satellite network without using any terrestrial resources. These advantages come along with a challenge; in contrast to geostationary (GEO) satellites, LEO satellites change their position with

CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF

329

reference to a fixed point on the Earth. Owing to this mobility, the coverage region of an LEO satellite is not stationary. A global coverage at any time is still possible if a certain number of orbits and satellites are used. The coverage area of a single satellite consists of small-sized cells, which are referred to as spotbeams. Different frequencies or codes are used in different spotbeams to achieve frequency reuse in the satellite coverage area.

Location management in the LEO satellite network environment represents more challenging problem because of the movement of satellite footprints. As a consequence, an LA cannot be associated with the coverage area of a satellite because of very fast movement of a LEO satellite. Thus, 4G will need the development of new LA definitions for satellite networks as well as the signaling issues mentioned for all of the location management protocols. In del Re [47], LAs are defined using (gateway, spotbeam) pairs. However, the very fast movement of the spotbeams results in excessive signaling for location updates. In Ananasso and Priscoli [73], LAs are defined using only gateways. However, the paging problem has not been addressed in the same reference.

Handoff management ensures that ongoing calls are not disrupted as a result of satellite movement, but rather transferred or handed off to new spotbeams or satellites when necessary. If a handoff is between two spotbeams served by the same satellite, handoff is intrasatellite. The small size of spotbeams causes frequent intrasatellite handoffs, which are also referred to as spotbeam handoffs [74]. If the handoff is between two satellites, it is referred to as intersatellite handoff. Another form of handoff occurs as a result of the change in the connectivity pattern of the network. Satellites near to polar regions turn off their links to other satellites in the neighboring orbits. Ongoing calls passing through these links need to be rerouted. This type of handoff is referred to as link handoff [59, 60]. Frequent link handoffs result in a high volume of signaling traffic. Moreover, some of the ongoing calls would be blocked during connection rerouting caused by link handoffs.

11.2 CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF

The handoff attempt rate in a cellular system depends on cell radius and mobile speed as well as other system parameters. As a result of limited resources, some fraction of handoff attempts will be unsuccessful. Some calls will be forced to terminate before message completion. In this section we discuss analytical models to investigate these effects and to examine the relationships between performance characteristics and system parameters. For these purposes some assumptions about the traffic nature are needed. We assume that the new call origination rate is uniformly distributed over the mobile service area with the average number of new call originations per second per unit area a. A very large population of mobiles is assumed, thus the average call origination rate is for practical purposes independent of the number of calls in progress. A hexagonal cell shape is assumed. The cell radius R for a hexagonal cell is defined as the maximum distance from the center of

a cell to the cell boundary. With the cell radius R, the average new call origination rate per

cell R is R = 3 3R2 a/2. Average handoff attempt rate per cell is Rh. The ratio γ0 of handoff attempt rate to new call origination rate (per cell) is γ0 Rh/R . If a fraction PB of new call origination is blocked and cleared from the system, the average rate at which new calls are carried is Rc = R (1 PB). Similarly, if a fraction Pfh of handoff attempts fails, the average rate at which handoff calls are carried is Rhc = Rh(1 Pfh). The ratio

330 MOBILITY MANAGEMENT

γc of the average carried handoff attempt rate to the average carried new call origination rate is defined as γc Rhc/ Rc = γ0(1 Pfh)/(1 PB).

The channel holding time TH in a cell is defined as the time duration between the instant that a channel is occupied by a call and the instant it is released by either completion of the call or a cell boundary crossing by the mobile. This is a function of system parameters such as cell size, speed and direction of mobiles, etc. To investigate the distribution of TH we let the random variable TM denote the message duration, that is, the time an assigned channel would be held if no handoff is required. The random variable TM is assumed to be

exponentially distributed T ( ) = μMeμMt with the mean value ¯M( 1M). The speed f M t T

in a cell is assumed to be uniformly distributed on the interval [0, Vmax].

When a mobile crosses a cell boundary, the model assumes that vehicular speed and direction change. The direction of travel is also assumed to be uniformly distributed and independent of speed. More sophisticated models would assume that the higher the speed the fewer changes in direction are possible.

The random variable Tn is the time a mobile resides in the cell to which the call is originated. The time that a mobile resides in the cell in which the call is handed off is denoted Th. The pdfs fTn(t) and fT h(t) will be discussed in Section 11.3.

When a call is originated in a cell and gets a channel, the call holds the channel until the call is completed in the cell or the mobile moves out of the cell. Therefore, the channel holding time THn is either the message duration TM or the time Tn for which the mobile resides in the cell, whichever is less. For a call that has been handed off successfully, the channel is held until the call is completed in the cell or the mobile moves out of the cell again before call completion.

Because of the memoryless property of the exponential distributions, the remaining message duration of a call after handoff has the same distribution as the message duration. In this case the channel holding time THh is either the remaining message duration TM or mobile residing time Th in the cell; whichever is less. The random variables THn and THh are therefore given by

THn = min(TM, Tn ) and THh = min(TM, Th)

The cumulative distribution functions (CDF) of THn and THh can be expressed as

FTHn (t) = FTM (t) + FTn (t)[1 FTM (t)]

FTHh (t) = FTM (t) + FTh (t)[1 FTM (t)]

The distribution of channel holding time can be written as

 

Rc

 

 

 

 

Rhc

 

 

FTn (t) =

 

 

FTHn

(t) + Rc + Rhc

FTHn

(t)

Rc + Rhc

 

1

 

 

 

 

γc

 

 

=

 

FTHn (t) +

 

FTHh (t)

 

 

1 + γc

1 + γc

 

 

= FTM (t) +

1

 

1 FTM (t) FTn (t) + γc FTn (t)

1 + γc

From the initial definitions,

FTH

(t)

 

1

eμMt

+

eμMt

F

(t)

+

γ

F

(t) ,

for t

0

=

 

 

1+γc

Tn

 

c

Th

 

 

 

 

 

0,

 

 

 

 

 

 

 

 

 

elsewhere

(11.1)

(11.2)

(11.3)

(11.4)

CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF

331

The complementary distribution function FC TH (t) is

FC TH (t) = 1 FTH (t) = FTH (t)

= 1 eμMt

eμMt

FTn (t) + γc FTh (t) ,

for t 0

(11.5)

1 + γc

0,

 

 

 

 

 

elsewhere

 

By differentiating Equation (11.4) we get the probability function (PDF) of TH as

 

fTH (t) = μMeμMt +

eμMt

 

μMeμMt

 

 

 

 

fTn (t) + γc fTh (t)

 

FTn (t) + γc FTh (t)

1 + γc

1 + γc

 

 

 

 

 

 

 

(11.6)

To simplify the analysis the distribution of TH is approximated in References [75, 76] by

a negative exponential distribution with mean ¯H ( 1H). From the family of negative

T

exponential distribution functions, a function which best fits the distribution of TH, by

comparing FC (t) and eμHt is chosen which is defined as

TH

μH min

μH

0

FC (t)

eμHt

dt

(11.7)

TH

 

 

 

Because a negative exponential distribution function is determined by its mean value, we

¯

 

 

 

 

 

 

choose TH( 1H), which satisfies the above condition. The ‘goodness of fit’ for this

approximation is measured by

 

 

 

 

 

 

 

FC (t)

eμHt

dt

 

 

G =

0

TH

 

 

(11.8)

 

 

 

 

 

2

FC (t) dt

 

 

 

 

0

TH

 

 

 

 

 

 

 

 

 

In the sequel the following definitions will be used:

(1)The probability that a new call does not enter service because of unavailability of channels is called the blocking probability, PB.

(2)The probability that a call is ultimately forced into termination (though not blocked) is PF. This represents the average fraction of new calls which are not blocked but which are eventually uncompleted.

(3)Pfh is the probability that a given handoff attempt fails. It represents the average fraction of handoff attempts that are unsuccessful.

(4)The probability PN that a new call that is not blocked will require at least one handoff before completion because of the mobile crossing the cell boundary is

 

PN = Pr {TM > Tn } =

1 FTM (t) fTn (t) dt = eμMt fTn (t) dt

(11.9)

0

0

 

332MOBILITY MANAGEMENT

(5)The probability PH that a call that has already been handed off successfully will require another handoff before completion is

PH = Pr {TM > Th} =

1 FTM (t) fTh (t) dt = eμMt fTh (t) dt (11.10)

0

0

Let the integer random variable K be the number of times that a nonblocked call is successfully handed off during its lifetime. The event that a mobile moves out of the mobile service area during the call will be ignored since the whole service area is much larger than the cell size. A nonblocked call will have exactly K successful handoffs if all of the following events occur:

(1)It is not completed in the cell in which it was first originated.

(2)It succeeds in the first handoff attempt.

(3)It requires and succeeds in k 1 additional handoffs.

(4)It is either completed before needing the next handoff or it is not completed but fails on the (k + 1)st handoff attempt.

The probability function for K is therefore given by

Pr{K = 0} = (1 PN) + PN Pfh

 

 

 

 

 

 

 

 

Pr{K = k} = PN(1 Pfh)(1 PH + PH Pfh){ PH(1 Pfh)}k1,

k = 1, 2, . . .

 

 

 

 

 

 

 

 

 

 

 

(11.11)

and the mean value of K is

 

 

 

 

 

 

 

 

 

 

K¯

 

k Pr

K

=

k

} =

 

PN(1 Pfh)

 

(11.12)

 

1 PH(1 Pfh)

 

= k=0

{

 

 

 

If the entire service area has M cells, the total average new call attempt rate which is not

 

¯

blocked is M Rc, and the total average handoff call attempt rate is KM Rc. If these traffic

¯

¯

components are equally distributed among cells, we have γc = (KM Rc)/(M Rc) K .

11.2.1 Channel assignment priority schemes

The probability of forced termination can be decreased by giving priority (for channels) to handoff attempts (over new call attempts). In this section, two priority schemes are described, and the expressions for PB and Pfh are derived. A subset of the channels allocated to a cell is to be exclusively used for handoff calls in both priority schemes. In the first priority scheme, a handoff call is terminated if no channel is immediately available in the target cell (channel reservation – CR handoffs). In the second priority scheme, the handoff call attempt is held in a queue until either a channel becomes available for it, or the received signal power level becomes lower than the receiver threshold level (channel reservation with queueing – CRQ handoffs).

11.2.2 Channel reservation – CR handoffs

Priority is given to handoff attempts by assigning Ch channels exclusively for handoff calls among the C channels in a cell. The remaining C Ch channels are shared by both new

 

CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF

333

ΛR + ΛRh

ΛR + ΛRh

ΛR + ΛRh

ΛRh

ΛRh

ΛRh

 

E

E

EE

 

E

E

 

0

1

CC-C-Chh

 

CC-C-Ch+1

C

 

μH

2μH

(C-CH)μΗ

(C-CH+1)μΗ

CμH

 

Figure 11.21 State-transition diagram for channel reservation – CR handoffs.

calls and handoff calls. A new call is blocked if the number of available channels in the cell is less than or equal to Ch when the call is originated. A handoff attempt is unsuccessful if no channel is available in the target cell. We assume that both new and handoff call attempts are generated according to a Poisson point process with mean rates per cell of R and Rh, respectively. As discussed previously, the channel holding time TH in a cell is approximated

¯

E j

of a

to have an exponential distribution with mean TH( 1H). We define the state

cell such that a total of j calls is in the progress for the base station of that cell. Let Pj represent the steady-state probability that the base station is in state E j ; the probabilities can be determined in the usual way for birth-death processes discussed in Chapter 6. The pertinent state-transition diagram is shown in Figure 11.21.

The state equations are

 

 

 

 

 

 

 

 

 

 

 

R + Rh P

j1

,

for

j

=

1, 2, . . . , C

C

h

 

jμH

 

 

 

 

 

 

Pj =

Rh

 

 

 

 

 

 

 

 

 

(11.13)

 

 

Pj1

,

 

 

for

j = C Ch + 1, . . . , C

 

jμH

 

 

As in Chapter 6, by using Equation (11.13) recursively, along with the normalization con-

dition Pj = 1, the probability distribution {Pj } is

j=0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P

CCh (

R

+

 

Rh

)k

C

(

R +

 

Rh

)CCh

k(CCh)

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rh

 

 

 

 

 

 

0 =

 

 

 

 

 

 

+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k=0

 

k!μHk

 

 

 

 

 

k!μHk

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k=CCh+1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

( R + Rh) j

P ,

 

 

for

j

=

1, 2, . . . , C

C

 

 

 

 

 

j!μ j

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

h

 

Pj =

 

 

 

H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(11.14)

 

( R + Rh)CCh Rjh(CCh)

P , for

j

=

C

C

h +

1, . . . , C

 

 

 

 

 

 

 

 

 

 

j!μ j

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The probability of blocking a new call is PB =

C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j=CCh Pj and the probability of handoff

attempt failure Pfh is the probability that the state number of the base station is equal to C. Thus Pfh = Pc.

11.2.3 Channel reservation with queueing – CRQ handoffs

When a mobile moves away from the base station, the received power generally decreases. When the received power gets lower than a handoff threshold level, the handoff procedure

334 MOBILITY MANAGEMENT

Forced terminations

Calls in progress

Queue of

Channel

delayed

use

Delayed

use

handoff

record

Handoff

record

attemps

 

Attemps

 

Delayed

Blocked

Handoff

New call

attempts

originators

Figure 11.22 Call flow diagram for channel reservation with queueing-CRQ handoffs.

is initiated. The handoff area is defined as the area in which the average received power level from the base station of a mobile receiver is between the handoff threshold level (upper bound ) and the receiver threshold level (lower bound ). If the handoff attempt finds all channels in the target cell occupied, we consider that it can be queued. If any channel is released while the mobile is in the handoff area, the next queued handoff attempt is accomplished successfully. If the received power level from the source cell’s base station falls below the receiver threshold level prior to the mobile being assigned a channel in the target cell, the call is forced into termination. When a channel is released in the cell, it is assigned to the next handoff call attempt waiting in the queue (if any). If more than one handoff call attempt is in the queue, the first-come-first-served queuing discipline is used. The prioritized queueing is also possible where the fast moving (fast signal level losing) users may have higher priority. We assume that the queue size at the base station is unlimited. Figure 11.22 shows a schematic representation of the flow of call attempts through a base station.

The time for which a mobile is in the handoff area depends on system parameters such as the speed and direction of mobile travel and the cell size. We call it the dwell time of a mobile in the handoff area TQ . For simplicity of analysis, we assume that this dwell time

is exponentially distributed with mean ¯Q ( 1H). We define j as the state of the base

T E

station when j is the sum of the number of channels being used in the cell and the number of handoff call attempts in the queue. For those states whose state number j is less than equal to C, the state transition relation is the same as for the CR scheme. Let X be the elapsed time from the instant a handoff attempt joins the queue to the first instant that a channel is released in the fully occupied target cell. For state numbers less than C, X is equal to zero. Otherwise, X is the minimum remaining holding time of those calls in progress in the fully occupied target cell. When a handoff attempt joins the queue for a given target cell, other handoff attempts may already be in the queue (each is associated with a particular mobile). When any of these first joined the queue, the time that it could remain on the queue without succeeding is denoted by TQ (according to our previous definition). Let Ti be the remaining dwell time for that attempt which is in the ith queue position when another handoff attempt

 

 

CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF

335

 

ΛR + ΛRh

ΛR + ΛRh

ΛR + ΛRh

ΛRh

ΛRh

ΛRh

ΛRh

 

E

0

E

EE

 

 

E

EC+1

 

 

E1 1

CC-C-Chh

 

CC

 

 

μH

2μH

(C-CH)μΗ

(C-CH+1)μΗ

CμH

CμH+μQ

CμH+2μH

 

Figure 11.23 State-transition diagram for CRQ priority scheme.

joins the queue. Under the memoryless assumptions here, the distributions of all Ti and TQ are identical. Let N (t) be the state number of the system at time t. From the description of this scheme and the properties of the exponential distribution it follows that

Pr {N (t + h) = C + k 1|N (t) = C + k}

 

= Pr {X h or T1 h or . . . Tk h}

 

= 1 Pr {X > h and T1 > h or . . . Tk > h}

(11.15)

= 1 Pr {X > h} Pr {T1 > h} . . . Pr {Tk > h} = 1 e(CμH +kμQ )h

since the random variables X, T1, T2, . . . , Tk are independent. From Equation (11.15) we see that it follows the birth-and-death process and the resulting state transition diagram is as shown in Figure 11.23.

As before, the probability distribution {Pj } is easily found to be

P

CCh

(

 

 

Rh

)k

C

(

R +

 

Rh

)CCh k(CCh )

 

=

 

 

 

 

R

+

 

 

 

 

+

 

 

 

 

 

 

 

 

Rh

 

 

 

 

 

 

0

 

 

 

 

k!μHk

 

 

 

 

 

 

 

 

 

k!μHk

 

 

 

 

 

 

 

 

 

 

k=0

 

 

 

 

 

 

 

 

 

 

k=CCh+1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+

 

 

( R + Rh)CCh kRh(CCh )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k C

CμH + iμQ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k=C+1 C!μCH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i=1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

( R + Rh) j

 

P ,

 

 

 

 

 

for

 

1

j

C

C

 

 

 

 

 

 

 

j!μ j

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

h

 

 

 

 

 

 

 

 

 

H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

( R + Rh)CCh Rjh(CCh )

P ,

 

for

 

C

C

h +

1

j

C

Pj

 

 

 

 

 

 

j!μHj

 

 

0

 

 

 

 

 

 

 

 

 

(11.16)

 

 

( R + Rh)(CCh ) Rjh(CCh )

P ,

 

for

 

j

C

+

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

j

C

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C!μCH

 

CμH + iμQ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i=1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The probability of blocking is PB =

Pj . A given handoff attempt that joins the

j=CCh

queue will be successful if both of the following events occur before the mobile moves out

336 MOBILITY MANAGEMENT

of the handoff area:

(1)All of the attempts that joined the queue earlier than the given attempt have been disposed.

(2)A channel becomes available when the given attempt is at the front of the queue.

Thus the probability of a handoff attempt failure can be calculated as the average fraction of handoff attempts whose mobiles leave the handoff area prior to their coming into the queue front position and getting a channel. Noting that arrivals that find k attempts in queue enter position k + 1, this can be expressed as

Pfh

(11.17)

PC+k Pfh|k

k=0

wherePfh|k = Pr {attempt fails given it enters the queue in position k 1}.

Since handoff success for those attempts which enter the queue in position k + 1 requires coming to the head of the queue and getting a channel, under the memoryless conditions assumed in this development, we have

1 Pfh|k =

k

 

P (i | i + 1) Pr get channel in first position}

(11.18)

 

i=1

 

whereP (i | i + 1) is the probability that an attempt in position i + 1 moves to position i before its mobile leaves the handoff area.

There are two possible outcomes for an attempt in position i + 1. It will either be cleared from the system or will advance in queue to the next (lower) position. It will advance if the remaining dwell time of its mobile exceeds either:

(1)at least one of the remaining dwell times Tj , j = 1, 2, . . . , i, for any attempt ahead of it in the queue; or

(2)the minimum remaining holding time X of those calls in progress in the target cell.

Thus

1 P(i | i + 1) = Pr {Ti+1 X, Ti+1 Tj , j = 1, 2, . . . , i} i = 1, 2, . . . .

 

(11.19)

1 P(i | i + 1) = Pr {Ti+1 X, Ti+1 T1, . . . , Ti+1 Ti }

 

= Pr {Ti+1 min(X, T1, T2, . . . , Ti )}

(11.19a)

= Pr {Ti+1 Yi } i = 1, 2, . . .

 

where Yi min(X, T1, T2, . . . , Ti ). Since the mobiles move independently of each other and of the channel holding times, the random variables, X, Tj , ( j = 1, 2, . . . , i) are statistically independent. Therefore, the cumulative distribution of Yi in Equation (11.19) can be written as

FYi (τ ) = 1 − {1 FX (τ )}{1 FT1 (τ )} . . . {1 FTi (τ )}

CELLULAR SYSTEMS WITH PRIORITIZED HANDOFF

337

Because of the exponentially distributed variables, this gives

FYi (τ ) = 1 eCμHτ eμQ τ . . . eμQ τ = 1 e(CμH+iμQ )τ

and Equation (11.19) becomes

1 P(i | i + 1) = Pr {Ti+1 Yi } = {1 FYi (τ )} fTi+1 (τ ) dτ

0

 

 

 

 

 

(11.20)

 

 

 

 

 

μQ

 

 

 

= e(CμH+iμQ )τ μQ eμQ τ dτ =

 

 

 

 

, i = 1, 2, . . .

Cμ

H +

(i

+

1)μ

 

0

 

 

 

Q

The handoff attempt at the head of the queue will get a channel (succeed) if its remaining dwell time T1 exceeds X. Thus

Pr {get channel in front position} = Pr {T1 > X} and

 

 

 

 

Pr {does not get channel in front position} = Pr {T1 X}

 

(11.21)

μQ

 

 

 

= eCμHτ μQ eμQ τ dτ =

 

 

 

 

Cμ

H +

μ

Q

0

 

 

The probability Equation (11.21) corresponds to letting i = 0 in Equation (11.20) Then from Equation (11.18) we have

1 Pfh|k =

k

 

 

 

Pr {get channel in first position}

 

 

 

P ( i| i + 1)

 

 

 

 

i=1

 

 

CμH + 2μQ

 

 

 

 

 

 

 

 

 

CμH + μQ

 

 

CμH + kμQ

 

 

CμH

 

(11.22)

=

CμH + 2μQ CμH + 3μQ

· · · CμH + (k + 1) μQ CμH +

 

μQ

=

CμH

 

 

 

 

 

 

 

 

 

 

 

 

 

CμH + (k + 1) μQ

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P

fh|k =

 

(k + 1) μQ

 

 

(11.23)

 

CμH + (k + 1) μQ

 

 

 

 

 

 

The above equations form a set of simultaneous nonlinear equations which can be solved for system variables when parameters are given. Beginning with an initial guess for the unknowns, the equations are solved numerically using the method of successive substitutions.

A call which is not blocked will be eventually forced into termination if it succeeds in each of the first (l 1) handoff attempts which it requires but fails on the lth. Therefore,

 

 

(1

 

 

)l1 Pl1

 

Pfh PN

(11.24)

P

P

P

P

=

 

 

1 PH (1 Pfh)

F = l=1

fh

n

 

fh

H

 

where PN and PH are the probabilities of handoff demand of new and handoff calls, as defined previously. Let Pnc denote the fraction of new call attempts that will not be completed because of either blocking or unsuccessful handoff. This is also an important system