Backoff strategies for demand re-registration in PCS database failure recovery

Backoff strategies for demand re-registration in PCS

database failure recovery

Yang Xiao*

Computer Science Division, Department of Mathematical Sciences, The University of Memphis, 373 Dunn Hall, Memphis, TN 38152, USA

Received 1 November 2002; revised 16 August 2003; accepted 2 October 2003

Abstract

Mobility Databases, Home Location Register (HLR) and Visitor Location Register (VLR), are utilized to support mobility management

for Mobile Stations (MSs) in Personal Communications Services (PCS) networks. If the location database fail, the subscribers’ services

will be seriously degraded due to the loss or corruption of location information. Previous work proposed demand re-registration with a p-

persistent backoff strategy and a checkpoint method that demonstrates better performance than the periodic re-registration policy. In demand

re-registration, after a VLR fails, it broadcasts a re-registration request to all MSs. Backoff strategies are needed since collisions may occur if

all MSs try to re-register after receiving the request. Choosing a good backoff strategy for demand re-registration has several benefits that are

our goals in this paper. A better strategy will save the re-registration traffic in terms of signaling cost. Moreover, a better strategy will allow an

MS to re-register earlier to reduce the probability that a call termination with expensive paging operations happens earlier than location

information recovery. In this paper, we propose and study seven backoff strategies for demand re-registration: one optimal p-persistent

strategy, three dynamic p-persistent strategies, and three non-persistent strategies. Among these proposed strategies, the optimal p-

persistent strategy is optimal in the sense of optimality among all the p-persistent strategies; the three dynamic p-persistent strategies improve

p-persistent strategies by allowing the p-value to change with time; the three non-persistent strategies include a binary exponential backoff

strategy, an exponential backoff strategy, and a non-exponential non-persistent backoff strategy; our studies show that they can be

approximately equivalent to special dynamic p-persistent strategies. Our studies also show that one of the dynamic p-persistent backoff

strategies is the best strategy among all the seven proposed strategies and our results indicate that with better backoff strategies, the

performance of demand re-registration can be dramatically improved.

q 2003 Elsevier B.V. All rights reserved.

Keywords: Database failure recovery; Personal Communications Services; Registration

1. Introduction

In a Personal Communications Services (PCS) network,

the service area is partitioned into several Location Areas

(LAs). Each LA consists of a number of cells. In each cell,

there is a Base Station (BS) with a number of Mobile

Stations (MSs). BSs in an LA are connected to a Mobile

Switching Center (MSC). One of the major tasks of mobility

management is to update MSs’ locations when MSs moves

from one place to another. The location update is also

referred to as registration procedure initialed by the MS.

Mobility Databases, Home Location Register (HLR) and

Visitor Location Register (VLR), are utilized to support

mobility management for MSs in PCS networks, and are

where the location information is stored. For every LA,

there is a VLR associated with it. When an MS visits the LA,

a temporary record of the MS is created in the VLR to

indicate its location. For every MS, there is a permanent

record stored in the HLR with the current information of

this MS.

Many papers have been proposed for location manage-

ments in PCS networks [1–7], but few papers pay attention

to fault-tolerance issues on location databases. If the

location database fail, the subscribers’ services will be

seriously degraded due to the loss or corruption of location

information. Afterwards, location record of an MS will be

automatically restored with or without some cost by the

registration event, the call origination event and the call

termination event. In other words, the VLR record is

Computer Communications 27 (2004) 400–411

www.elsevier.com/locate/comcom

0140-3664/$ - see front matter q 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.comcom.2003.10.002

* Tel.: þ1-901-678-2487; fax: þ1-901-678-2480.

E-mail address: [email protected] (Y. Xiao).

http://www.elsevier.com/locate/comcom

recovered if the MS follows a normal registration procedure

without any cost; if the MS originate a call referred to call

origination procedure, the VLR record cannot be found, the

request is rejected, the MS then is asked to initial a location

registration procedure, and the location information is

restored with some cost; if another user (wired or mobile)

call the MS before the location information is recovered, a

call termination procedure [8] needs to be followed with

expensive paging procedure to all LAs in order to find the

MS. Therefore, it is desirable to recover the location

information with registration procedures before call termin-

ation and call origination events happen to avoid expensive

paging cost and signaling cost.

In Ref. [9], a periodic re-registration was proposed in

which the MSs periodically re-register to the VLR. With an

appropriately chosen frequency of location re-registration

studied in Ref. [10], there is a better chance that periodic re-

registration recovers the location information before a call

termination arrives, and at the mean time, reduces the

signaling cost. In Ref. [11], a demand re-registration was

proposed with a p-persistent backoff strategy and a

checkpoint method that was shown to outperform periodic

re-registration. In demand re-registration, after a VLR fails,

it broadcasts a re-registration request to all MSs. Since

collisions may occur if all MSs try to re-register after

receiving the request, a p-persistent backoff strategy is

adopted for MS re-registrations in Ref. [11]. The perform-

ance is good but not when the number of MSs is large.

Therefore, a checkpoint method was proposed to improve

the performance. In the checkpoint method, the number of

MSs entering a VLR is counted when the VLR is in normal

operation. When the counted number is larger than a

threshold, these records are check-pointed into a backup.

When the VLR fails, most of the records can be restored

from backups, i.e. the records of MSs who came before the

last checkpoint time, and only records between the last

checkpoint time and the VLR failure need to re-register. The

number of MSs needed to re-register is smaller than the

threshold. Therefore, the number of MSs needed to re-

register can be kept small. As stated in above, this approach

outperforms periodic re-registration with less cost.

Choosing a better backoff strategy for demand re-

registration has several benefits. It will save the re-

registration traffic in terms of signaling cost. Moreover, it

will allow an MS to re-register earlier to reduce the

probability that a call termination with expensive paging

operations happens earlier than the location information

recovery. These benefits are our goals of this study. In this

paper, we propose and study seven backoff strategies for

demand re-registration: one optimal p-persistent strategy,

three dynamic p-persistent strategies, and three non-

persistent strategies. In the p-persistent strategy, we propose

an algorithm to find the optimal p-value. The optimality is

the sense of optimality among all p-persistent strategies. In

the proposed dynamic p-persistent strategies, we lose that

fact the p-value is constant all the time in p-persistent

strategies so that p-value is dynamic changed with time. The

idea is inspired by the fact that each MS who plans to re-

register has only one message to send, i.e. after a successful

registration, the number of MSs who plan to re-register

decreases. Therefore, the p-value should be a non-increas-

ing function of time. For the three non-persistent strategies,

we propose and study a binary exponential backoff strategy,

an exponential backoff strategy and a non-exponential non-

persistent backoff strategy. We prove that a non-persistent

strategy can be approximately equivalence to a special

dynamic p-persistent strategy.

The rest of the paper is organized as follows. In Section

2, we extend the demand re-registration with p-persistent

strategy [11] by losing the fact the p-value is a constant. In

Section 3, we propose and study several backoff strategies

mentioned above. Performance study is presented in Section

4. Finally, we conclude our paper at Section 5. Throughout

this paper, an ‘active’ MS stands for an MS who plans to re-

register but did not successfully re-register yet after a VLR

failure. On the other hand, a ‘non-active’ MS stands for an

MS who has already successfully re-registered.

2. Backoff demand re-registration

In Ref. [11], a p-persistent backoff strategy was adopted

for the demand re-registration. In other words, p is a

constant number. Such a strategy doe not consider the

following fact: after a VLR failure, each MS has only one

re-registration message to send. Therefore, after a success-

ful re-registration, the number of active stations decreases,

and p should be non-increasing function of time instead of

being constant. In this section, we lose this assumption so

that p can be, but not necessarily be, constant.

In existing TDMA-based mobile systems such as

GSM, message delivery through the radio interface is

performed in timeslots. We assume that the re-registration

message if transmitted can finish in one timeslot. We also

assume that there are N MSs in the VLR at the moment

when the VLR fails.

After a VLR failure, the VLR broadcasts the re-

registration request to MSs. After an MS receives the

request, it will send re-registration message with probability

p1 at timeslot 1. If collided, it will send the message again

with probability p2 at timeslot 2, so on so forth until it

successfully transmits the re-registration message. In

general, at timeslot i; if it has not successfully transmitted

its re-registration message yet, it will send the message with

the probability pi: In other words, {p1; p2; p3;…; pt;…} are

used for transmission probabilities until the MS successfully

transmits its re-registration message. We consider some

special cases of above general form as follows. If pi is a

constant value for all the timeslots, i.e. pi ¼ a for i $ 1; it is

a p-persistent backoff strategy, which was adopted in Ref.

[11]. If pi is not constant, we call such a strategy as a

dynamic p-persistent strategy. Another kind of backoff

Y. Xiao / Computer Communications 27 (2004) 400–411 401

strategies is non-persistent backoff strategies in which,

before an MS decides transmit its re-registration message, it

first backoffs some timeslots, and then transmits. The

backoff time is a uniformly chosen integer number from an

interval ½0; b�; and b is increased if collided. As derivations

in Section 3, a non-persistent backoff strategy can be

approximated by a dynamic p-persistent backoff

strategy. Therefore, p-persistent, non-persistent and

dynamic p-persistent all can be treated as special cases of

the general form that we are presenting here.

Let pðnÞ be the probability that an MS x successfully

transmits its re-registration message at timeslot n: We

have

pð1Þ ¼ p1ð1 2 p1ÞN21 ð1Þ

In order to derive pðnÞ; let Xn denote the number of MSs

except x have successfully sent the re-registration messages

up to timeslots n: Therefore, P½Xn ¼ k� is the probability

that MS x has not completed the re-registration up to

timeslot n; and k MSs had already successfully sent their re-

registration messages. Xn forms a discrete time Markov

chain, shown in Fig. 1, where n $ 0 and 0 # k # minðn;NÞ:

Note that when we are talking timeslot n; we mean the time

till the end time of timeslot n; and for n ¼ 0; it is just for

normalized purpose.

As shown in Fig. 1, if the current state is k at timeslot n

where 0 # k # minðn;NÞ; the next state will be either the

same ðkÞ or changed to k þ 1 at timeslot n þ 1 depending on

whether there is a successful re-registration or not in the

current slot. Therefore, we have following iterative

equations

P½X0 ¼ 0� ¼ 1 ð2Þ

P½Xn ¼ 0� ¼P½Xn21 ¼ 0� 12N

1

!pnð12pnÞ

N21

" #ð3Þ

P½Xn ¼ k� ¼P½Xn21

¼ k21�N 2 k

1

!pnð12pnÞ

N2k

" #þP½Xn21

¼ k� 12N 2 k

1

!pnð12pnÞ

N2k21

" #for 0, k

,minðn;NÞ ð4Þ

P½Xn ¼k� ¼P½Xn21 ¼ k21�

�N 2 k

1

0@

1Apnð12pnÞ

N2k

24

35; for k

¼minðn;NÞ ð5Þ

The first part of Eq. (4) is derived by considering the case

that the MS x does not successfully re-register with a

probability 12pn at timeslot n; and the second part of Eq.

(4) is derived by considering the case that the MS x

successfully re-registers with a probability pn at timeslot n:

As shown in Fig. 1, with Eqs. (2)–(5), we can derive the

probability at any state and at any timeslot.

We will derive pðnÞ with P½Xn ¼ k� as follows. i MSs

except x have re-registered up to timeslot n 2 1 with

probability P½Xn21 ¼ i 2 1�; and the MS x successfully re-

registers at timeslot n with probability pnð1 2 pnÞN2i21:

Therefore, we have

pðnÞ ¼Xminðn;NÞ21

i¼0

P½Xn21 ¼ i 2 1�pnð1 2 pnÞN2i21 ð6Þ

Let EðnÞ be the average number of elapsed timeslots

before an MS has successfully re-registered, and it indicates

the elapsed time between the time when the VLR fails and

Fig. 1. Transition diagram of Xn with respect to time.

Y. Xiao / Computer Communications 27 (2004) 400–411402

the time when demand re-registration is successfully

performed for the MS. We have

EðnÞ ¼X1n¼1

npðnÞ ð7Þ

Assume that the normal registrations of an MS form a

Poisson process, and the time period between the VLR

failure and the next registration has an exponential

distribution. The time period can be represented by a

geometric distribution with probability g1 since the

events in the mobile system happens at timeslots.

Similarly, the times for the first call origination and the

first call termination can be represented by geometric

distributions with probabilities g2 and b; respectively,

assuming that the call originations and call terminations

are Poisson processes. The normal registration traffic and

the call origination traffic can be aggregated to form a

geometric distribution with probability g ¼ g1 þ g2: The

values of N;g;b can be obtained from OA&M of a

mobile system [12].

After a VLR failure, let t1; t2; t3 be the times when the

MS re-registration occurs, the first MS normal registration

or call origination occurs, and the MS call termination

occurs, respectively. If minðt1; t2Þ , t3; the expensive

paging operation for the MS call termination can be avoided

without demand re-registration. Therefore, demand re-

registration is effective when t1 , t3 # t2; but not when

minðt1; t2Þ , t3: Let P1;d ¼ P½t1 , t3 # t2� be the

probability that demand re-registration is effective, and

P2;d ¼ P½minðt1; t2Þ , t3� be the probability that demand

re-registration is not effective. We have

P1;d ¼ P½t1 , t3 # t2�

¼X1n¼1

PðnÞX1

j¼nþ1

bð1 2 bÞ j21X1i¼j

gð1 2 gÞi21

24

35

8<:

9=;

¼X1n¼1

pðnÞbð1 2 bÞnð1 2 gÞn

bþ g2 bg

� �ð8Þ

P2;d ¼ P½minðt1; t2Þ , t3� ¼ 1 2 P½t3 # t1; t3 # t2�

¼ 1 2X1n¼1

pðnÞXn

j¼1

bð1 2 bÞj21X1i¼j

gð1 2 gÞi21

24

35

8<:

9=;

¼ 1 2X1n¼1

pðnÞb½1 2 ð1 2 bÞnð1 2 gÞn�

bþ g2 bg

� �ð9Þ

Let u ¼ P1;d=P2;d be the demand re-registration’s contri-

bution of saving expensive paging. We will use u and EðnÞ

as measures to study the performance of demand re-

registration.

3. Backoff strategies

In this section we propose seven backoff strategies: one

optimal p-persistent strategy in Section 3.1, three dynamic

p-persistent strategies in Section 3.2, and three non-

persistent strategies in Section 3.3. The optimal p-persist-

ent is a p-persistent strategy with an optimal p value. The

three dynamic p-persistent strategies include a simple

dynamic p-persistent strategy, and optimal dynamic

p-persistent strategy, and a measurement-based dynamic

p-persistent strategy. The three non-persistent strategies

include one binary exponential backoff strategy, an

exponential backoff strategy and one non-exponential

non-persistent backoff strategy.

3.1. Optimal p-persistent backoff (O-PPB) strategy

In this subsection, we consider the p-persistent backoff

strategy, a special case of dynamic p-persistent backoff

strategies when pi is a constant value for all the timeslots,

i.e. pi ¼ a for i $ 1: This is the case proposed in

Ref. [11] where the effect of a on the EðnÞ was analyzed.

In this subsection, we propose a method to find the

optimal a value.

As the observations in Ref. [11], we analyze the effect of

a on EðnÞ shown in Fig. 2. EðnÞ is a decreasing function of a

when a is small until a reaches a point, and then EðnÞ

becomes an increasing function of a: The reasons are stated

as follows. When a is small, there are a lot of empty slots

wasted, the number of collisions is quite small, and

therefore EðnÞ is a decreasing function of a: When a is

large, there are a lot of collisions, the empty slots wasted

becomes not important, and therefore EðnÞ becomes an

increasing function of a: Let the value amin provide the

minimum EðnÞ: The value amin could be derived by

differentiating EðnÞ with respect to a and let the result be

zero. However, we have not a closed form equation for EðnÞ:

Another approach might work according to the following

fact. The minimum EðnÞ is reached when the empty slots

wasted is balanced with the collision slots. In other words,

the average number of empty slots equals the average

number of collision slots. However, both the average

number of empty slots and the average number of collision

slots has not closed-forms. Therefore, it is very complex to

derive a closed-form of amin; if not impossible. Here, we

adopt a simple search algorithm as follows to approximate

the value amin that provides the minimum EðnÞ: Let d be a

pre-defined precision value. In other words, the difference

between the result of the following algorithm and amin

should be smaller than d:

Step 1. Initially, let L ¼ 0 and R ¼ 1; and they stand for

the left endpoint and right endpoint of the search interval

½L;R�:

Step 2. If R 2 L # d; let amin ¼ L þ R=2; and stop.

Step 3. Let len ¼ ðR 2 LÞ=4:


Step 4. Calculate the values of EðnÞ when

a ¼ L;L þ len;L þ 2len;L þ 3len;R: Let EL;ELþlen;

ELþ2len;ELþ3len;ER stand for these values, respectively.

Compare the values among them.

Step 5. If one of the smallest values is ELþlen; let L ¼ L

and R ¼ L þ 3len; and go step 2.

Step 6. If one of the smallest values is ELþ3len; let

L ¼ L þ len and R ¼ R; and go step 2.

Step 7. Let L ¼ L þ len and R ¼ L þ 3len; and go step 2.

To explain the effectiveness of the above algorithm, we

have the following facts. When observing EðnÞ’s curve

shape in Fig. 2 with the above explanations, we know that

amin will never be L or R all the time. Our purpose here is to

always keep amin within the interval ðL;RÞ and the interval

becomes smaller after each step until the length of the

interval is smaller than d:

Since amin is related to N; a practical usage of this

algorithm is stated as follows. For each reasonable N;

calculate amin offline using the above algorithm. Then, a

table of N and amin is created and used online.

3.2. Dynamic p-persistent backoff strategies

In this section, we consider the dynamic p-persistent

backoff strategies, and try to fully utilize the fact that the

number of active MSs is not increasing function with

respect to time. We first introduce a simple dynamic

p-persistent backoff strategy. Then, we refine it to be the

optimal dynamic p-persistent strategy. Finally, we

introduce a measurement-based dynamic p-persistent

strategy.

3.2.1. Simple dynamic p-persistent backoff (SD-PPB)

Let xt be the random variable to stand for the number of

active stations before timeslot t: The probability that there is

a successful transmission at timeslot t is

Psuccess;t ¼xt

1

!ptð1 2 ptÞ

xt21 ¼ xtptð1 2 ptÞxt21 ð10Þ

In order to save the re-registration traffic, Psuccess;t should

be as large as possible. In order to get the maximum value of

Psuccess;t; we differentiate it with respect to pt; and let it

equals to zero.

›Psuccess;t

›pt

¼ xtð1 2 ptÞxt21 2 xtptðxt 2 1Þð1 2 ptÞ

xt22

¼ xtð1 2 ptÞxt22½1 2 ptxt� ¼ 0 ð11Þ

Then, we have pt ¼ 1=xt which may provide Psuccess;t the

maximum value, the minimum value, or none of them. We

will prove that it provides Psuccess;t the maximum value as

follows. We have the following facts

(1) When pt ¼ 0 or pt ¼ 1; we have Psuccess;t ¼ 0;

(2) We have Psuccess;t $ 0 when pt [ ½0; 1�;

(3) Psuccess;t is a continuous function of pt;

(4) 1=xt is the only point to make ›Psuccess;t=›pt ¼ 0 in the

interval (0,1).

Based above facts and the knowledge of Calculus, we

know that pt ¼ 1=xt provides Psuccess;t the maximum value.

With this conclusion in mind, we define the Simple

Dynamic P-Persistent Backoff (SD-PPB) strategy as

Fig. 2. EðnÞ vs. a when N ¼ 100:


follows.

p1 ¼1

N; p2 ¼

1

N 2 1;…; pN21 ¼

1

2; pN ¼ 1; and

pi ¼ 1; for i . N: ð12Þ

We can formally re-write them as

pi ¼

1

N 2 ði 2 1Þ; if 1 # i # N

1; if i . N

8><>: ð13Þ

However, the above algorithm optimistically believes

that there is always a successful transmission at the previous

timeslot since 1=xt provides a maximum value of Psuccess;t;

and ignores that there is still a small probability that there is

not any successful transmission in the previous timeslot.

Therefore, SD-PPB is not the optimal backoff strategy, but it

tries to approximate the optimal one. We will lose this

assumption in the next strategy, which is the analytic

optimal strategy.

3.2.2. Optimal dynamic p-persistent backoff (OD-PPB)

In this section, we improve the SD-PPB strategy by the

following observations. We have p1 ¼ 1=N in the first

timeslot. Let EðxtÞ be the mean of xt: To derive p2; let r1

be the probability that there is not any successful

transmission in the first timeslot. r1 ¼ 1 2 Np1ð1 2

p1ÞN ¼ 1 2 ð1 2 1=NÞN : Then, the mean number of

stations Eðx2Þ after the first timeslot will be Eðx2Þ ¼

r1N þ ð1 2 r1ÞðN 2 1Þ instead of ðN 2 1Þ: Therefore, we

should define p2 ¼ 1=½r1N þ ð1 2 r1ÞðN 2 1Þ� instead

of 1=ðN 2 1Þ: In later timeslots, the similar analysis

also holds.

Formally with EðxtÞ; we re-write Psuccess;t in Section 3.2.1

as follows

Psuccess;t ¼EðxtÞ

1

0@

1Aptð12 ptÞ

EðxtÞ21 ¼ EðxtÞptð12 ptÞEðxtÞ21

ð14Þ

Similar to Section 3.2.1, pt ¼ 1=EðxtÞ provides the

maximum value for Psuccess;t: To derive EðxtÞ; let ri be the

probability that there are no successful transmissions during

timeslot i: We have

Eðx1Þ ¼ N ð15Þ

p1 ¼ 1=Eðx1Þ ð16Þ

r1 ¼ 1 2 Eðx1Þp1ð1 2 p1ÞEðx1Þ ð17Þ

In general, we have following iterative equations for

t . 1

EðxtÞ ¼ rt21Eðxt21Þ þ ð1 2 rt21ÞðEðxt21Þ2 1Þ ð18Þ

pt ¼ 1=EðxtÞ ð19Þ

rt ¼ 1 2 EðxtÞptð1 2 ptÞEðxtÞ ð20Þ

From Eqs. (15)–(17) and iterative Eqs. (18)–(20), we

can derive all values of {p1; p2; p3;…; pt;…}:

3.2.3. Optimal measurement-based dynamic p-persistent

backoff (OMBD-PPB)

In this subsection, we assume that the MSs can sense the

channel, and have a clear knowledge of their and others’

transmission results: successful or failed. Therefore, based

on the OD-PPB strategy, we propose the following

measurement-based (by observations) optimal dynamic p-

persistent backoff strategy. We can treat this strategy as a

measurement-based version of the OD-PPB strategy. We

have

p1 ¼ · · · ¼ pl121 ¼1

N; ð21Þ

pl1¼ · · · ¼ pl221 ¼

1

N 2 1; ð22Þ

pl2¼ · · · ¼ pl321 ¼

1

N 2 2; ð23Þ

· · ·

plN22¼ · · · ¼ plN2121 ¼

1

2; ð24Þ

plN21¼ · · · ¼ plN21 ¼ 1; ð25Þ

where l1; l2;…; lN are the timeslots before which a

successful transmission happens, and this fact is known by

observing the channel.

The idea of the OMBD-PPB strategy is the same as the

OD-PPB strategy except that in the OMBD-PPB strategy

MSs know a successful transmission by observations;

whereas in the OD-PPB strategy, MSs know a successful

transmission by analytical calculations. Intuitively, under

the assumption that an MS can know its and others’

transmission results, this strategy is more practical than that

in the OD-PPB strategy, especially in a noisy environment.

3.3. Non-persistent backoff strategies

In this subsection, we consider non-persistent backoff

strategies. In a non-persistent backoff strategy, an MS

executes the following steps until successful transmission of

its re-registration message.

Step 1 i ¼ 0

Step 2. Backoff a number of slots that is randomly chosen

from {0;…;Bi 2 1}; where Bi is the backoff window size

for ith iteration.

Step 3. Transmit.

Step 4. If collided, let i ¼ i þ 1 and go to step 2,

otherwise, stop.


B0 is normally pre-defined and is called the initial

backoff window size. For an MS, during the first

backoff stage, it will randomly choose a timeslot among

{0;…;B0 2 1}: The probability that it chooses any timeslot

among {0;…;B0 2 1} to transmit is 1=B0 since we consider

the behavior of the MS on average. On average, it will

transmit at timeslot B0=2: Therefore, we have

p1 ¼ · · · ¼ pðB0=2Þ¼

1

B0

; ð26Þ

pðB0=2Þþ1 ¼ · · · ¼ pðB0=2ÞþðB1=2Þ¼

1

B1

; ð27Þ

In general, we have

pXi21

k¼0

ðBk=2Þþ1

¼ · · ·¼ pXi

k¼0

Bk=2

¼1

Bi

; i¼ 1;…;m;…

ð28Þ

Therefore, we proved in above that a non-persistent

backoff strategy is approximately equivalence to a dynamic

p-persistent backoff strategy. We need to determine further

the backoff window sizes {B0;B1;…Bi;…}: How to choose

the backoff sizes defines different non-persistent backoff

strategies.

Binary exponential non-persistent backoff strategies

(BE-NPB) are very popular non-persistent backoff strat-

egies in many protocols such as the IEEE 802.3

(Ethernet). In the binary exponential backoff strategy, we

have Bi ¼ 2Bi21; i $ 1: Other exponential non-persistent

backoff strategies (E-NPB) are defined as Bi ¼ aBi21; i $ 1:

For both BE-BPB and E-BNP, by adjusting the initial

window size B0; we can have different dynamic p-persistent

backoff strategies. In other words, B0 is a tunable parameter

in BE-NPB and E-NPB. Even though it is difficult to find the

optimal B0; an advantage of BE-NPB/E-NPB over non-

dynamic/dynamic p-persistent backoff strategies is that it

does not required to know the number of MSs, and it can

adjust its backoff stages automatically On the other hand, for

non-dynamic/dynamic p-persistent backoff strategies

requires the knowledge of number of MSs to achieve better

performance.

Even though exponential backoff strategies are popular,

it is difficult to find the optimal B0: Therefore, we propose a

non-exponential non-persistent backoff strategy (NE-NPB)

as follows. We set Bi equal the number of active MSs at the

beginning of the backoff stage i in order to let everyone

success at this backoff stage, intuitively. After the MS failed

to re-register, a new backoff stage begins iteratively. In

other words, the backoff window size equals to the number

of active MSs. Therefore,

B0 ¼ N; ð29Þ

For i ¼ 1; 2;…;m;…; we can derive Bi through Bi21

iteratively. An MS will begin its next stage at Bi21=2 þ 1 on

average. At the beginning of timeslot Bi21=2 þ 1;

the average number of active MSs is derived by following

iterative procedure. Let Cj be the probability that there is a

successful transmission at timeslot j þ ki ðj ¼ 0;…;Bi21=2Þ;

and Di is the number of active MSs at the beginning of

timeslot j þ ki ðj ¼ 0;…;Bi21=2 þ 1Þ; where ki is the first

timeslot at the beginning of this stage i: We have following

iterative procedure to derive Bi through Bi21:

D0 ¼ Bi21 ð30Þ

For j ¼ 0;…;Bi21=2; we iterate following

Cj ¼ Dj 1 21

Bi21

� �Dj21 1

Bi21

� �; ð31Þ

Djþ1 ¼ CjDj þ ð1 2 CjÞðDj 2 1Þ; ð32Þ

Finally, we have

Bi ¼ DBi21=2þ1 ð33Þ

By Eqs. (29) – (33), we can derive all Bi;

i ¼ 1; 2;…;m;…; iteratively.

4. Performance evaluation

In this section, we will study the performance of the

proposed backoff strategies for demand re-registration. We

will use u and EðnÞ as performance metrics to evaluate these

strategies. In Section 4.1, we will show the proposed

Optimal P-Persistent Backoff (O-PPB) Strategy is optimal

among all the p-persistent strategies. In Section 4.2, we

compare the performance of the three dynamic p-persistent

strategies. In Section 4.3, we compare the three non-

persistent strategies. In Section 4.4, we compare the p-

persistent, dynamic p-persistent and non-persistent strat-

egies. Finally, we discuss the tradeoffs of all the strategies in

Section 4.5

4.1. Optimality of O-PPB

Let d ¼ 0:0001; b ¼ 0:0005; and g ¼ 0:001: Then, we

exercise O-PPB proposed in Section 3, and we get following

results.

amin ¼0:0153; if N¼1000:0075; if N¼2000:0049; if N¼300

�ð34Þ

In order to verify our results are optimal among p-

persistent strategies, we draw Fig. 3 to show EðnÞ over a;

and Fig. 4 to show u over a: As shown in Fig. 3, a ¼ 0:0153;

0.0075, and 0.0049 give the minimum values of EðnÞ for

N ¼ 100; 200, and 300, respectively. As shown in Fig. 4,

a ¼ 0:0153; 0.0075, and 0.0049 give the maximum values

of u for N ¼ 100; 200, and 300, respectively. Also as shown

in Fig. 3, EðnÞ increases as N increases. As shown in Fig. 4,

u decreases as N increases.


Fig. 5 shows u over N for the optimal-p-persistent

strategy and other p-persistent strategies. As illustrated,

the optimal-p-persistent is the best strategy among others.

The interesting phenomenon observed from the figure is

that the curve of the optimal-p-persistent is an envelop

of all other curves, and this envelop is the upper bound

of all other curves. This phenomenon shows optimality of

the optimal-p-persistent strategy among other p-persistent

strategies.

4.2. Comparisons of dynamics p-persistent strategies

Fig. 6 shows the values of u over N for the optimal-

dynamic p-persistent strategy and the simple-dynamic

Fig. 3. EðnÞ vs. a:

Fig. 4. u vs. a:


p-persistent strategy. It is clearly that the optimal-dynamic

p-persistent strategy is much better than the simple-

dynamic p-persistent strategy. In other words, even though

the optimal-dynamic p-persistent strategy also tries to be

approximate the optimal one, there is a large approxi-

mation error. The reasons are stated in the previous

sections.

4.3. Comparisons of non-persistent strategies

Fig. 7 shows the values of u over N for binary exponential

backoff strategies when the initial window sizes are 2, 4, 8,

16, 32, 128 and N: As shown in the figure, the results are

quite not sensitive to the initial window size when the initial

window sizes are 2, 4, 8, 16, 32, and 128. However, when

Fig. 5. p-persistent strategies.

Fig. 6. Dynamic p-persistent strategies.


N . 400; the result, when the initial window size is N; is

better than those when the initial window sizes are 2, 4, 8,

16, 32, and 128. The reason is that the initial window size is

chosen to be too small. Therefore, the knowledge of the

number of active stations helps a little bit to determine

the initial window size range. Note that the result when the

initial window size is N; is not the best case all the time, e.g.

when N ¼ 100; it is smaller than any of those when the

initial window sizes are 2, 4, 8, 16, 32, and 128.

Fig. 8 shows the values of u over N for exponential

backoff strategies when the bases are 2 (binary), 3, 4, and 5.

The initial window size is 8. As illustrated, the performance

decreases as the base increases, and the binary exponential

backoff strategy is the best among these four strategies.

Fig. 7. Binary exponential backoff strategies.

Fig. 8. Exponential backoff strategies.


Fig. 9 shows the values of u over N for the NE-NPB

strategy and the binary exponential backoff strategy. As

illustrated, the binary exponential backoff strategy outper-

forms the NE-NPB strategy.

4.4. Comparisons of O-PPB, OD-PPB, and BE-NPB

Fig. 10 shows the values of u over N for O-PPB, OD-

PPB, and BE-NPB. As illustrated, the optimal dynamic p-

persistent backoff strategy is the best strategy.

4.5. Discussion

The optimal p-persistent strategy is the best strategy

among all the p-persistent strategy. The optimal dynamic p-

persistent strategy is the best among all the strategies

proposed, and it is much less complex than the algorithm to

find the optimal p-persistent strategy.

An advantage of BE-NPB/E-NPB over non-dynamic/dy-

namic p-persistent backoff strategies is that it does not

required to know the number of MSs, and it can adjust its

Fig. 9. NE-NPB and the binary exponential.

Fig. 10. Comparisons.


backoff stages automatically On the other hand, for non-

dynamic/dynamic p-persistent backoff strategies requires

the knowledge of number of MSs to achieve better

performance.

5. Conclusion

This paper proposed and studied seven backoff strat-

egies for demand re-registration: one optimal p-persistent

strategy, three dynamic p-persistent strategies, and three

non-persistent strategies. Among these proposed strategies,

the optimal p-persistent strategy is optimal in the sense

of optimality among all the p-persistent strategies; the

three dynamic p-persistent strategies improve p-persistent

strategies by allowing the p-value to change with time;

the three non-persistent strategies include a binary

exponential backoff strategy, an exponential backoff

strategy and a non-exponential non-persistent backoff

strategy, and our studies show they can be approximately

equivalent to special dynamic p-persistent strategies. Our

study demonstrated that demand re-registration may

effectively recover VLR failure and the following results

were observed:

† Our algorithm to find the optimal p-persistent strategy

can effectively find the optimal one, which is the best

strategy among all p-persistent strategies. The curve of

the optimal p-persistent is an envelop of all other curves

of p-persistent strategies, and this envelop is the upper

bound of all other curves.

† The optimal-dynamic p-persistent strategy is much

better than the simple-dynamic p-persistent strategy.

† For non-persistent backoff strategies, the results are

quite not sensitive to the initial window size when

the difference between the initial window size and the

number of active stations are smaller than 400. The

number 400 comes from our results and therefore it can

be treated as a constraint.

† For exponential backoff strategies, the performance

decreases as the base increases, and the binary

exponential backoff strategy is the best among other

four strategies.

† The optimal dynamic p-persistent backoff strategy is the

best strategy among all the seven proposed strategies,

and our results indicate that with better backoff

strategies, the performance of demand re-registration

can be dramatically improved.

Acknowledgements

The author would like to thank Dr Yi-Bing Lin for

providing programs in Ref. [11] and giving many

comments.

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March (1997).

Yang Xiao received his PhD degree in computer science and

engineering from Wright State University, Dayton, Ohio, USA. He

had been a software engineer, a senior software engineer, and a

technical lead working in the computer industry from 1991 to 1996.

From 1996 to 2001, he had been awarded the DAGSI PhD Fellowship.

From August 2001 to August 2002, he worked at Micro Linear as a

MAC architect involving the IEEE 802.11 standard enhancement work.

Since August 2002, he has been an assistant professor of computer

science at The University of Memphis. He is a TPC member for

conferences: IEEE WCNC 2004, IEEE ICOS 2004, and IEEE ICCCN

2003. Heathcom 2003, and SCI 2003. He serves a co-chair in

Symposium on Data Base Management in Wireless Network

Environments in IEEE VTC 2003—Fall, and an associate guest editor

for International Journal of High Performance Computing and

Networking. He is a voting member of the IEEE 802.11 Working

Group, and a member of IEEE and ACM. His current research interests

include WLANs, WPANs, and PCS.


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