Directed Retry Blocking Paper

8/6/2019 Directed Retry Blocking Paper

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ANALYTICAL MODEL FOR EVALUATING BLOCKING IN GSM

NETWORKS WITH DIRECTED RETRY

Arnie Neidhardt, Judith Jerkins, and K. R. Krishnan

Telcordia Technologies, New Jersey, U.S.A.

E-mail: {arnie, jlj, krk}@research.telcordia.com

ABSTRACT

We present an analytical model and propose an efficient algorithm for evaluating call blocking

in GSM networks with directed retry, where a call blocked by the sector of its initial attempt

may be redirected to one or more alternative sectors. Since exact solution by Markov-chain

analysis is unfeasible in large networks owing to the size of the state-space, we propose a method

for an approximate solution, which is shown to be highly accurate in two small examples.

.

1. Introduction

In this paper, we propose a method for evaluating

call blocking in GSM networks with Directed Retry,

in which a call blocked by the sector1

of its initial

attempt may be redirected to one or more alternative

sectors that are also accessible to the call. The

purpose of directed retry is to increase the effective

capacity of a network by making it possible for calls

to be served by the frequencies of multiple sectors (so

long as the frequency assignments in sectors reduces

co-channel interference and provides calls with low-

interference access to multiple sectors).

We present an analytical model and propose an

efficient computational method for estimating

blocking in networks with directed retry, which can

be used, in particular, to compare network

performance for different frequency assignments. An

efficient analytical algorithm for performance

evaluation is attractive, even if its results are only

approximate, since the alternative of using detailed

simulations can be a very time-consuming process.

The general problem of estimating blocking in

networks employing Directed Retry has been

addressed in previous literature [1-4]. While themethods proposed in all these papers are based on the

assumption that blockings in different sectors are

independent, a distinctive feature of our method is

that such independence is not assumed. A uniform

loading in all cells is assumed in [1,4], and only one

overflow attempt for calls eligible for directed retry is

actually considered in [1,3], though the possibility of

extension to attempts on multiple overflow paths is

mentioned. The analysis in [2] accounts for the

possibility of more than one overflow path with

directed retry but imposes a given retry sequence on a

1In this paper, the term sector is used to mean a base station andits primary group of frequencies, and, occasionally, also the

geographic area served primarily by the base station.

specified fraction of all call attempts in a cell,including already-redirected calls. Our method can be

applied in arbitrary finite networks, and conforms to

the specified attempt-sequences for calls eligible for

directed retry.

The papers [1-4] above consider handoff calls

explicitly, which, like the fresh calls, are also

assumed to constitute a Poisson arrival process of

known rate. It follows that, with that assumption,

handoff calls can be taken into account by redefining

the source traffic to include such handoff calls aswell. This is the point of view adopted here.

In our analysis, the network is visualized as a set

of non-overlapping geographical bins (e.g., a grid).

We assume we are given the call-origination load foreach bin, and the sequence of primary and overflow

servers that provide service to those calls. The call-

origination load is defined as the traffic load imposed

on the network by calls initiated by users situated in

the geographical region of the bin, as well as by

existing calls that were handed off as the user enteredthe bin. The server-sequence, in general, can

correspond to servers in the primary and overflow

sectors of directed retry that serve the bin, ranked in

decreasing order of signal clarity for calls originatingin the bin. We assume that the concept of Directed

Retry applies to both initial calls and handoffs. Theobjective is to determine the blocking experienced by

calls originating or handed-off in each bin, as well as

the overall network blocking.

The overall problem can be viewed as one where a

set of traffic sources (users in the bins) is served, each by a given subset of servers, attempted in a given

sequence of primary and overflow servers, with the

different sources possibly limited to different subsets

of servers. Thus, the problem corresponds to a

generalized limited-availability system, special formsof which have been investigated in the telephony

literature [5]. The exact solution of such problems by


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the analysis of state-evolution of Markov chains is

rendered unfeasible in realistic-sized networks by the

explosion in the number of possible states. Therefore,we propose a method that allows us to obtain an

approximate solution by solving a system of fixed-

point equations whose solution is easily obtained by

means of an iterative procedure that is guaranteed toconverge.

We wish to arrive at a description of network load

due to call-segments. It is clear that the average

segment duration is shorter than the average sessionduration, but the average initiation rate for segments

is higher than the average initiation rate for sessions

(since a single session can spawn several segments

during its life). The magnitude of the overall load,which is the product of an initiation rate and an

average duration, is independent of whether the ratesand durations refer to segments or to sessions. For the

situation being considered here, the relevant

magnitude of demand is the product of the segment-

initiation rate and average segment duration. In

particular, the traffic demand for each segment will

be assigned to the bin in which the phone is located atthe time the segment is initiated. To allow for the

specification of a well-posed problem without having

to specify the mobility patterns of users, the

assumption will be made that, during a period of

interest for performance (such as a busy hour), thedemand for call-segments takes the form of a Poisson

process, exactly as assumed in [1-4]. In the remainder

of the paper, we use the terms call and call-segment

interchangeably, to mean call-segment.

In Section 2, we introduce the notation needed for

a mathematical description of the problem, and

proceed to develop the general algorithm for

evaluation of call blocking, on the basis of certain

approximations that ensure tractability of the

computations for networks of realistic size. In Section3, we illustrate the application of our method in two

examples, for which, in fact, exact solutions by

Markov chain analysis is also practical because of the

limited number of possible states. In the first example

of a 2-sector network with directed retry, whichinvolves 64 states, our method is shown to achieveexcellent accuracy in the determination of the

capacity increase made possible by directed retry,

with a relative error that is smaller than about 1.25%

over a wide range of overlap between the sectors. For

a symmetric 3-sector network, with 1202

states, the

relative error remains within 3.3% over a range of

overlap among sectors. Section 4 summarizes our

paper and suggests the desirability of applying the

method to realistic-sized cellular networks.

2.2. Server Teams

A call that seeks a traffic channel (time-slot) from

a sector can be served by any available traffic channel

of any frequency in the sector. Therefore, we regard

the combined group of traffic channels corresponding

to all the frequencies in a sector as an indivisible

server-team, presenting itself as an indivisible unit

(which we will sometimes refer to as an atom) to allcalls that seek service from it. This indivisible server-

team corresponding to a sector can be thought of as a

full-access trunk-group in the general teletraffic

parlance, with the number of trunks equal to the

total number of traffic channels in the frequencies

serving the sector.

2.

Mathematical Model for Service withDirected Retry

2.1.Calls and Call-SegmentsFor a user on a mobile phone, a single

communication session may require the support of a

succession of servers as the user traverses a path

through the domain of the network. From the users

perspective, the one long session is a single call, but

from the perspective of the network, the shortersegments of the session (corresponding to different

traffic channels, i.e., servers, being used to support

different segments of the session upon each handoff)

represent different demands on the network. Each

segment represents an occasion on which the network

might refuse to carry the call. Here we view all

occasions of not serving the call (whether in the

initial segment or in handoffs) with equal concern,

and so we shall treat each segment of a

communications session as a separate call; with this

language, every call not served initially, and every

call dropped during handoff, will be described as a

blocked call3.

2.3. Notation

The primary notation used for our analysis is given

here.Let

=T the set of all sectors in the network

=tS the indivisible server-team (atom) for sector

Tt =B the set of bins in the network

=bl the rate of call-segment originations in bin

b (assumed to be a Poisson process)

=bt the mean duration of call-segments

originating in bin b

bbba tl=

=bL the sequence (ordered list) of indivisible2 The 3-sector problem considered has 512 states, but with the

assumption of symmetry, this is reduced to 120.

3 A call not served in its initial segment would typically be

designated a blocked" call, while a call that is not served in a later

segment would typically be designated a call dropped during

handoff.


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server-teams that a call originating in bin b can

attempt

=UL the unordered set of the elements in any

server-team sequence L

=bN =)( bLN the total number of servers in bL

=

)(LN the total number of servers in anyserver-team sequence L

X = the number of elements of the set X

(ordered or unordered)

The aim of the investigation is to estimate , the

blocking probability for call-segments originating in

bin , . In the course of constructing an

algorithm for this purpose, we shall also make

estimates of , the load offered to the indivisible

server-team

bb

b Bb

tz

tSt T, .

As stated earlier, the server-team of a sectorpresents itself as an indivisible trunk-group to all calls

that seek service from it. Hence, , the sequence of

server-teams accessible to a call-segment originating

in bin b , can be viewed as a sequence of trunk-

groups. Call-segments originating in bin b attemptthe trunk-groups in the specified sequence, but are

free to access any idle trunk within each trunk-group.

bL

A Caveat on Notation

The analysis in this paper proceeds on the basis of

several approximations that we introduce in order to

achieve an efficient algorithm for a computational

solution. It is to be understood that the variousquantities we introduce and calculate are, in fact,

estimates of their true values, derived on the basis

of the approximations. In order to avoid introducing

burdensome new notation to distinguish each such

estimate from the corresponding true value, we shall,

in fact, re-use the notation introduced for a true valuealso for its estimate that appears in our calculations,

depending on the context to make our meaning clear.

Notation applied to Two-Sector Example

To illustrate the use of the notation we have

introduced, consider the simple two-sector network

example of Figure 1. The two sectors have a region ofmutual overlap, such that calls originating within the

region of overlap can be served by the servers of both

sectors, while calls originating in the non-overlap

region of each sector are served only by the servers of

that sector. Let the combined Poisson offered load of

the two sectors be erlangs, out of which a fractionAarises from the region of overlap. The frequencies

in Sector 1 correspond to an indivisible server-team

with servers and those in Sector 2 correspond

to an indivisible server-team with servers.

1S 1s

2S 2s

For simplicity, we assume symmetry in the

distribution of demand. Of the portion of

the total load that originates outside the overlap

region, we take

)1( gA -

2

)1( gA -

( 1s +

1S

2S

1S

bLbb },,

}

to arise from each sector,

being served only by the servers of that sector. The

load , originating in the region of overlap, is

served by a team of servers. However, a

mobile in the overlap region does recognize a

difference in signal clarity between the frequencies of

the two sectors, and its call is set up on a frequency ofthe clearer sector if a traffic channel is free, otherwise

it is attempted on the other sector. Thus, half of the

overlap load comes from calls that first attempt

the server-team , and then, if blocked, overflow to

the other server-team , while the other half of the

load is due to calls that attempt the server-teams in

the sequence .

Ag

,bb

)2s

B

Ag

2S

,t

B

=

b

b

b

b

b

=

bnet

b

b

t=b b

=

b

bnet

a

}bt

}

lb

This two-sector network may be viewed as

consisting of four bins, as shown in Figure 1, with

the pattern of offered loads and server-sequences

shown in Table 1. We will return to this example after

an exposition of our performance evaluation

algorithm in the next Section.

2.4.Performance Evaluation

Problem Statement

Given{ , we wish to determine

the blocking for the call-segments of the individualbins{ , and the overall network blocking

, given by

bl

b

netb

b

b

b

bb

b

a

a

t

t

b

l

l

(1)

If it is reasonable to work with the approximationthat the mean holding times are the same for all bins,

i.e., t for all , then we have

b

bb

a

b

b (2)

This is the case we consider in this paper, so that

the analysis can proceed from knowledge of only the

bin-loads{ , without the need for the separate

specification of{ and{ .

ba

}


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g

Bin 4Bin 1 Bin 2 Bin 3

Figure 1: Four Bins in the 2-Sector Network

Bin b Offered Load ba Server Sequence bL

12

)1(1

gAa

-

=

1S

22

2

Aga =

21SS

32

3

Aga =

12SS

42

)1(4

gAa

-

= 2S

TABLE 1: Bin Structure of 2-Sector Network

Motivation for an Approximation in the Solution

In principle, the exact solution to the problem

framed above can be obtained by solving the

fundamental state-transition equations of the

underlying Markov chain. However, since trunk-group

has states, the total number of states, given

by , becomes astronomically large for

realistic-sized networks, and the exact solution defies

computation.

tS

t

[ 1+ts[ +ts 1

]]

T

For this reason, we seek a simpler, approximate

formulation in which we express blockings of sets of

servers (i.e., server sets) in terms of equivalent loads

offered to them. In particular, there is the general case

of a set of limited-access trunk-groups, in which

different classes of calls might have their access limited

to just subsets of the trunk-groups. Various instances oflimited-access servers known as graded multiples

have been studied in the literature of telephony [5]. The

approach we present can be thought of as a method of

obtaining an approximate solution to the general

problem that includes graded multiples.

The Relevance of the Load on the Server-team

Sequence bLWhether a specific call-segment originating in a bin

gets served depends only on whether or not all the

servers of the server-team sequence are

occupied at the time that call-segment is offered,

regardless of the order in which the servers are

attempted. However, the probability that a call

originating in bin is blocked could be influenced by

the server-team sequences of all the bins, because those

sequences affect the total load imposed on the

servers in . This dependence arises from the

possibility of cascades of blocking, when blocking atone team of servers in one location leads to the

b

bN bL

b

bN

bL


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redirection of some load to another team at another

location, which, in turn, might block and redirect other

load to yet another team. t

bt

S

bf

team-serverthetooffered

isthatbinofloadofferedoffraction=.

Now, , then . If , we write

, where is the initial server-sequence

appearing before the appearance of in , and is

the remainder of the sequence after the appearance of

. In this case, the fraction of load that is offered

to in is the fraction that is blocked by the set of

servers in the initial sequence . Since the

number of servers is

bt LS if

bttRS

bL

UbtI

0=btf bt LS

t bL

ba

btI

btb IL =

tS

tS

btI

S btR

btI , we write

We will approximate this complicated dependence

of the blocking probability of bin b on all the

redirected loads by calculating the blocking probability

when a single effective Poisson load Y is offered to

the unordered set of servers . Thus, we estimate

by the following Erlang-B formula (the

appropriateness of the use of the Erlang-B formula will

be discussed later):

bb

U

bL

b

bb

)(,(U

btbtbt IYIBf = , (5),(3).( bbb YNB ,=b )

System of Equations for Determining Loadswhere stands for the yet-to-be-defined load

offered to the set of servers .

)(U

btIY

UbtI

We turn now to the derivation of a system ofequations from which the loads { in (3), associated

with a particular bin b and offered to the set of

servers , can be obtained. This will be a system of

}bY

U

bL

T equations, where T

=

is the number of sector

server-teams, each team being an indivisible atom (full-

access trunk-group) to all calls seeking service from it.

In Section 2.3, these server-team atoms were denoted

by , with the load offered toTtSt, tz TtSt,

In the special case where itself is the first element

of , ; this special case can be included in the

above notation by defining, in this case, and

further defining Y .

tS

(B

bL 1=btf

,btI

1)0,0,0)(

We thus have

( )

=

bt LSb

Ubtbtbt IYIBaz

:

)(, , (6).Tt

In deriving the equations for { , we find that we

also have to consider loads other than only the { .}bY

}bY

For , the right hand side of the system of

equations (6) involves, in general, offered loads on

various unordered sets of servers V , which are

unspecified as yet. We next specify these loads in Step2.

Tt

UbtI=

Load on Server-Team Atoms tS

The equations for determining the load Y on the

unordered set of server-teams are derived in two

steps.

b

U

bL Step 2: The Specification of Y in terms of{ )(V }tz

Consider a general unordered set V of server-team

atoms, say,nttt

SSS ,,21K=V .

Step 1: Equations for { }tz

In the first step, we write equations for the loads

offered to each of the atoms . The

formulation of equations for these atomic loadsrequires us, in general, also to deal with the concept of

the load Y offered to a limited-access set Vof

atoms, in which access for different calls could be

limited to different subsets of the component atoms.

TtSt,

)(V

If , i.e., if V consists of a single server-team,

say, , then, and is among the

loads to be determined by solution of (6).

1=n

tS tt zSYY == )(V)(

When , it is possible that different subsets of

serve different groups of calls, perhaps with no calls

given full access to all the servers in . Various

instances of such limited-access service systems have

been studied in the early literature of telephony [5]. Weintroduce the following definition of offered load forsuch a system:

2n

V

V

We begin by writing

(4),

=

Bb

bbtt afz

whereFor a general set V of two or more atoms

(server-teams), a bin-load is considered

offered to V if an atom of V appears as the

ba


6/9


7/9

practical terms, if one halts the iterative calculation at a

finite stage k at which the differences are

satisfyingly small, then the bounds show

that the solution is basically identified to within ones

desired accuracy. Since we have no proof of

uniqueness in general, cases of multiple solutions mayexist, and in such cases, the differences would

decrease to limits that are not all zero. In any case, ourproposed calculations can determine whether or not thesolution is unique for a network with given loads and

server-sequences.

kt

kt z-

ktt

ktz-

kt zz

kt

2)

),(

1

22

Az

zs

+

bL

}bb

,(

2

1sB

BAg

+

A

A

( 1

2

s

a

4

3

2

1

=

=

=

=

),(

),(

),(

),(

224

213

212

111

zsB

AssB

AssB

zsB

=

+=

+=

=

b

b

b

b

and

[ ]

+

++-

==

),(

),(),(2

)1(

21

2211

AssgB

zsBzsBg

a

a

b

b

b

bb

net

B

B

b

b

Comparison with Exact Solution

3. Examples: Comparison of Model with

Exact Solutions

For the case of two symmetrical sectors, each with

servers, the exact solution can be obtained by

analysis of the Markov chain that describes theevolution of the occupancy-vector of the two server-

teams . The number of possible states

is , and well-established numerical

methods were used to determine the equilibrium stateprobabilities.

7=s

(1+s

),( 21 SS

)(12+s 64)1 =

3.1.Symmetrical Two-sector Network

We now apply the performance evaluation algorithmdeveloped in Section 2 to the simple two-sector

network example that was described earlier at the endof Section 2.3. For this example (for which the loads

and server-sequences are summarized in Table 1), anexact solution is, in fact, feasible, and available forcomparison with the results produced by our method. Instead of making the comparison in terms of the

individual bin blockings in the two solutions, we

compare the capacity that is obtained in the 2-sectornetwork by having enabled a portion of the traffic to beserved by server-teams of both sectors. In particular,

under the performance constraint , we

estimate the total network load in erlangs that canbe served, for various values of the degree of overlap

01.0netb

A

g.

Offered Loads on Server-Team Atoms

For this example, 4,2 == BT , and Equations (6)

take the form:

2),

2),(

43122

22311

AgaazBaz

AzsBaaz

=++=

=++=

We present in Table 2 a comparison of the exact

solution with the approximate solution proposed here.For reference, we also include the trivial linearapproximation consisting of mere interpolation

between the known values at

and are readily solved by the iterative methoddescribed earlier.

Offered Load Y to Server-Team Sequenceb cells)t(coinciden1andoverlap)(no0 == gg .

Since we haveThe trivially interpolated result, overestimates the

capacity, and is subject to a maximum error of 7.5%.

24

123

212

11

SL

SSL

SSL

SL

=

=

=

=

We note that the approximation proposed in this

paper is slightly conservative, i.e., slightlyunderestimates the capacity, but is highly accurate: forthe cases considered, the relative error never exceeds

1.25%, and the mean relative error is only about 0.55%.

we obtain,

2

4321

4321

1

zY

aaaaY

aaaaY

zY

=+++

=+++

3.2.Symmetrical 3-Sector NetworkWe also carried out similar calculations for a

symmetric 3-sector network, with 7 servers in eachsector. Such a 3-sector network may be viewed as

consisting of 15 bins. For a range of sector overlaps,we again calculated the percentage error in theestimated total offered load that could be supported at a

1% blocking level. The degree of mutual overlap was

described by the fraction of calls in the network thathad access to only their home sector, to only two

sectors, and to all three sectors. The number of states in

from which we obtain the bin-blockings { and the

network blocking as follows:netb


8/9

a corresponding Markov chain would be 512; however, because we assumed symmetry, we could perform anexact Markov-chain analysis with only 120 states.

Leaving out the details of the formulas andcalculations in the interest of space, we found that the

maximum relative error between the exact solution andthe approximation furnished by our method was lessthan 3.3%.

4. Summary

We have presented an analytical model forestimating the performance of GSM cellular networksthat employ directed retry, in which a call blocked by

the sector of its initial attempt may be redirected to analternative sector that is also accessible to the call,thereby increasing the effective capacity of the

network. In essence, the problem is that of performance

evaluation of a generalized limited-availability system[5], special forms of which have been investigated in

the telephony literature. Since the exact solution ofsuch problems by Markov chain analysis is renderedimpractical by the explosion in the number of possible

states, we have proposed a method that allows us toobtain an approximate solution by solving a system offixed-point equations by means of an iterative

procedure that is guaranteed to converge. Unlikeprevious methods reported in the literature, our methoddoes not assume that blockings in different sectors are

independent. It proceeds by associating an offeredload not only with each sector but also with the set of

servers in each sequence of sectors that corresponds toa permitted retry-sequence in the network, by definingthe notion of an equivalent offered load for a generallimited-availability system of servers in which traffic

streams may have only partial access to sub-groups ofservers. The performance evaluation algorithm can beused to compare GSM network performance and

capacity for different frequency assignment plans.

We have presented a comparison for two simple

symmetric networks for which the exact solution byMarkov chain analysis is also available. In a simple 2-

sector network, we have shown that our methodproduces excellent accuracy in the determination of thecapacity made possible by directed retry, with a relativeerror that is smaller than about 1.25% over a widerange of overlap between sectors. In the case of asymmetric 3-sector network, the relative error from the

exact solution is less than 3.3% over the range ofmutual overlap considered amongst sectors.

While such accuracy is reassuring, such simpleexamples are no substitute for testing the validity ofour approximations and the accuracy of our

calculations against actual measurements of performance from realistic-sized cellular networks,with due account taken of the inherent statisticalvariability in the measurements. Obtaining such datafor comparison from commercial cellular networks presents its own challenges. We hope that the

possibility of simple analytical methods of

characterizing network performance will spur interestin applying our method to actual networks.

REFERENCES

1. B. Eklundh, Channel Utilisation and BlockingProbability in a Cellular Mobile Telephone System

with Directed Retry, IEEE Trans on Comm., vol.COM-34, no.4, April 1986, pp.329-337.2. D. McMillan, Traffic Modelling and Analysis forCellular Mobile Networks, Proceedings, 13thInternational Teletraffic Congress, pp.627-632,Copenhagen (Elsevier, North Holland), 1991.

3. Tak-Shing Peter Yum and Kwan Lawrence Yeung,Blocking and Handoff Performance Analysis ofDirected Retry in Cellular Mobile Systems, IEEETransactions on Vehicular Technology, Vol.44, No. 3,August 1995, pp. 645-650.4. R. Verdone and A. Zanella, "Analytical Evaluationof Blocking Probability in a Mobile Radio System withDirected Retry", IEEE Journal on Selected Areas inComm., pp. 322-331, Vol. 19 No. 2, February 2001.5. R. Syski, Introduction to Congestion Theory inTelephone Systems, Second Edition, Elsevier SciencePublishers B.V., Amsterdam, 1986.


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g

Exact

Solution

(E)

Proposed

Method

(M)

Linear

Interpolation

(L)

%Error in

100*

-

E

ME

%Error in L

100*

-

E

LE

0.0 5.00188 5.00188 5.00188 0.00 0.00

0.1 5.11292 5.10827 5.23686 0.09 -2.42

0.2 5.23910 5.22852 5.47184 0.20 -4.44

0.3 5.38400 5.36582 5.70682 0.34 -6.00

0.4 5.55228 5.52434 5.9418 0.50 -7.02

0.5 5.74981 5.70951 6.17678 0.70 -7.43*

0.6 5.98360 5.92842 6.41176 0.92 -7.16

0.7 6.26071 6.19001 6.64674 1.13 -6.17

0.8 6.58539 6.50520 6.88172 1.22* -4.50

0.9 6.95406 6.88686 7.1167 0.97 -2.34

1.0 7.35168 7.35168 7.35168 0.00 0.00

TABLE 2: Traffic Capacity of Two Symmetrical 7-Server Sectors

Total Offered Load in Erlangs for Overall Blocking of 1%

*Maximum relative error for method

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