BSTJ 60: 8. October 1981: Cutoff Calls and Telephone Equipment Reliability. (Tortorella, M.)

13.2 MB·English

Cutoff Calls and Telephone Equipment
Reliability

By M, TORTORELLA
(teruncit receives Maren St, 1981)

kis often cificul or expensive to measure culoff eas, which are
esl caused By fidures wel malfonctians in some compet of
the telephone network. Therefore, itis deirable to hasw an indirect
Imethod for estimating Use nasnber of eutoffcalis caused by euipmnent
(hilures in a stctehing sytem or factity. This paper dineumen a
Imuthematical madel thal can be used ta determine the cutoff eal! rate
Tuencneork component as a function of the fore nesies and faire
rates in the component, andl the call holding tine distribution, It
Includes a discussion of w paradign: for developing reliability objec.
tes thot directly reflect service an i is seen Dy end sera The
mathematical model. an SM/24jefe qraruing mate with serer fal
lures is desrided. & atvong lon of large numbers anda eentrat limit
theornm forthe number of clef ealls—eccumutated either acrurding
to the numberof failures ar mer tine~are developed. An example
rom eattehint system i given to show hots tne results are applied
fn apecifie cases

|. INTRODUCTION AND SUMMARY

‘The purpose ofthis papar ito daseribe u mothematical model for
tho raul raf calle caused by fur xe mlfunetions in telephone
feauipment. "The cutoff call bchavior uf wee any piece of telephone
‘equipment thst serv valle can be analyzed using this technique,
hut ehe primary applicalione we have in mind are rg inlays
‘patems contoining muy component, uch ax switching ayetenss and
{unsyssion sjstania (crunk groupe), ‘The model relates the roto of
cutofl calls posi by faut in tho oquipment nd ita subsystems
‘tothe fake mods in the equipment, ther eeverty and requeney af
fecurenee, ad the cll-bnlding-time cstrivoion

"The interaction of leleplone ell equests ch service equipment
Inns oftea beeamuccrsally deweribed uring quming madele Therefore,
‘tacoma reasonable thar a scudy of the effects of equipment failure on
{he casa celephone sytem should he feasible within the context of
the clasical queuing models of telephony. This és the approach
fadnpied here, ith the adtitonal feamure that the nervers may he
‘unreliable ane sujet. te flare «kind tbat cause the eaorer (i
tne in serve al api whose server fll be rope fo he
‘ymtam at che time of the failure, Forya and Meseeni have previoualy
sendiad unk groupe containing unreliable servers * Their interes: waa
in chavactersing the effect om arriving ells of one oe more hoe
holding-time (hence, very likely co be malfunctioning) cronka in the
‘grup, whereas here the intereat ia primerilyin the effec of unreliable
servers that may fil sngly oF together in groupe, on eustamern whe
‘really i srs.

"The puner i divide inv fivenections, Section T contains a recersl
<iscussion of riabilty objectives as they apply to telephone eat
‘ment, and the paradigm for developing reliability objectives that
Alrctly reflec. sprvice aa i fean by’ che cuscome. We observe that
the critical step that han been lacking is the ability to translate
equipment reliabilcy into rates of occurrence and duration of cus
tomerperceivable problems, auch an eutaf calla and nerwark connac

ona ofa queuing syetom with acrver
aes, und, asin Us nructurs, ch probiy Ut meal in Ose
‘eater wil le cutoff fx deleraine, ‘The way one descr rather
‘ealialy Ube aystera organization aed fallure modes i lao covered in
‘this rection. ‘The probability of cutof can he computed onder quite
feneral condiciona on the arrival proces, che service times, and the
queue discipline, because it depeads only on whal happens afler the

‘Bectin TY deacrihes a more speeiatied queuing, mel, in the
‘context of which certain Lint laws forthe eumslative numberof eatofT
falls oan be obtained, ‘Thief the M/M/e blacking spatem with oerver
failures und both ustrony la and a contra iit cheorem are obtaimed
‘The eventual use of thaso limit In, aa the basis for constructing
atistcal esis for deloreining complince with objectives, is also
Twefly discussed. Sotion V is devoted to the sngle-rerver ease, and
explci calculation of el parameters of interest.

Finally, Seotion VI gies an weample of Ue application of thi heory
to the estimation of cule! ell rats iw wll ewivehing systom. Tit
{important tv be rhe to do this kind of analysis heonee one many
to predict ew call performance for % aystera that i etl being
esigned. This lechrigue is thea un exumple of an indiret, albeit

1062 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBE 1984
‘approximate, method of estimating « cutoff eal rate for which mo
‘satisfactory dreet method may be avaiable.

‘Two appendices contain all proofs and other mathematical details
that, otherwise placed, would interfere withthe flow of the ext.

1 RELIABILITY ORJECTIVES AND CUSTOMER SERVICE
2.41 General

Tis currently recognized that the most desirable way to specify
performance and service objectives for telephone network equipment
4s ta use, in dition t economic information, considerations of how
the operation of thi equipment affoctsscrvice as it is seen by the
customer, In order 10 do chis for relishility objectives, we need co
‘elie that customers do not peresive outages, failures, and malfunc-
tions as such, They ara aware of them only insofar as they ean
fervice problems detactable by ers who generally are not wware of
the internal operations ofthe telephone network. Te achieve Une goal
of delermining equipment reliability objecaives based on eustomer
‘eed and expectations, then, the following steps are require:

{@) Determine the customer-perceivable service affct ofthe reli

bizy problems 'o be controlled

(i) Determine the quantialive relationshipa between the fre-
quency an duration ofveliabilily problema inthe system or equlpynent
fd the rates of agourrense and duration of hs service effecs found
{the iat step

ii) Use Uneserlotinahips to tranalals Use customer service sje:
‘ives forthe system, which euniol the customer-percevable eerte
Memuning fom rbiabty probewns, ino inceral eiaility objectives

"This panerfocunen on the second wie for a particule service effec:
cutoff alle,

2.2 Serica tects

failure nd malfunction are cuoffealis, ineffective atom
connection failures), jslation ine and toll) and rarer fmspaie-
iments (excessive low, noise, et}. Cutof? calls will be discussed at
Tenth below, Ineffective attempt, oF network eonneetion failures, can
‘pe eaused by fulures and malfunctions beeause the waavalablity of a
portion of the Iclephone network increases the network’ blocking
probability during the time tie portion af he networks out of service,
If Ge filed equipment is a customer’ loop, ot a part of tho local
central office that dibles the costomer’s fine Functions, causing
tustomer to be unable to communicate with the local central affice,
the customer experiences ine iation forthe duration of the allure.

CCUIOFF GALLS AND RELIABILITY. 1865
1f the failed equipment isa tall-connecting trunk group fom a cur-
tomer’s local central office, the customer experiences toll isolation,
‘meaning that toll cals to or from certain arees eannot be placod ot
received.

‘Transmission impairments can be ened by malfunctions auch a
cquipmont operating outside tolerances. ‘These phenomena are well

indertond and measstemenc plana are in place to return elevant
Jafurmation shout trnamiesion prablems te maintenance forces £0
‘har abnormal condicions may be corrected, These will not be discussed
ferther

"The rte of network connection faluzes aun dhe duration of eal:
tions are determined primarily by the duration ofthe outage, Thus,
[nalyis ofthese service: preblems is help in deter elihliy
‘objectives and maincanance palicias to limit outage duration. Wl
‘00 below that che rat of cutoff cll is primary driven by the race of
failures, so that analvas of cotoff alls i useful mainly in determining
objective for frequency of oocarrenes of outages, Of eoureo, a compre
Dbensve strategy for veliabilty manegement should deal with thea
complementary facets of equipment rlabity inn unified way, and
‘uintenance (service restoration and equipment repaz) policies play

i important part here. Aw abjertive fr feqiency of onsurrance of
Thilures, together wth» maintenance poliey, ieee cera Una
outage ime forthe equipment. Similuty, an objective for uta oulage
time, loether witha series reioraton nd equipmenLnaieatraegy,
Tite the numer af times outage may ccue.Allhough this paper
‘eae oly with cunft ealls and frequency af eosurrenee nf ostagem, it
Should e bore in mind that a united appenach to relay objec
tives, combining considerations nat only of ew cally and ou
frequency but also of network connection files and outage duration,
i most desirable.

2.3 Types of failures included

2.3.1 Causes of cutot one

‘A culo cal is # connected stable) call that has been verminaved
other than by an on-book by cithor party. The event of tamination is
sometimes roferrod to we & cuof) for short (as is call that is 60
sect). ‘The lerminology sintered (a connote an witentional,
‘nexpected interruption. IaLeraaional Telegraph sed Telephone Con
sultetve Commitee (ccrrs) terminology refer to a evtoffemusing
failure in a switching syacem as a “premature release malfunction in
an exchange.”

Cutoff ealis are eaused by equipment failures (including renovery
sctions}, and other extemal factors, wich radio files and in-band
(allot? (dznulation of Che 2600 Fi wuperviory signal by a signal emi

1884. THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1981
ted by one of Une paris). "The termination Lakes place atthe instant
the failure or uther even: egina, so the rale uf culos is influenced
primarily by the rate of falures (this is demonstrated in eq, (7)
CCatota are related eo reliability, chen, jst ms ineffective ettmpts or
network connection failures are related to availability. To determine
the rate of cutoff calls seen by a tclephone user, che eutoltcall
performances of inlvidual awitching an Lransmisiion systoms are
Fombined ina network soda. A autable model sone forthe reliability
(of seri aster consisting of avitching systoms and trunk groups,

12.32 Scope of the mode!
‘Tho reliably problems covered hy the model are thowe of failure
and ropuir af etre aystems and pars of systems, and those alles
tnd malfunctions that may noe completely dvable a aye or subsys
term, bu that cutoff calls when they occur. Th the first ease, epstems
tnd eubayetoms will be considered to be ether operating properly and
available for wie, or not operating at all and unavailable. Cutoff
falls caused by improper operation, or operation outside tolerances, of
{eyatom or subtethcam alo be tread. The key notion is that any
veut that causes encoff all when i cea can he ealld a “failure”
for purposes of ths discussion, The medel can accommodate many
ferent “failure” moder, at long we the occurrence times au weverities
of theee events can be characterised auMiciendly well chat failure
Droceses and culolf pacts (Seetion 82) can be assigned. In pie
Tur the adel gould im principle include ouch evenes n= rin fe
nT inehand valkoff we “falorr moder” Howove, in studying eutolt
falls a related to equipment reliability, this ie not recommended,
‘Devaue theta are ex events, not caused by ant euipment failure
Crmalfunction which cnuld be controlled by preventive or corrective
texion by the telephone company.

"As for cause of fale, forthe modal cher i no vestrietion un the
‘cause nf the failure or malfenction. ALL that i required i that one be
blo to lis. the nda of evens Unt cause eutofe, and describe proba
bilsially Mh cies boeween inciewa For each lind of event. The
scope ofthis wae encompasses all fallores which lad (v cutot cll
Tepardloss of cate, including bardware (component fairs), whware
tl Fra fae, man intervention errora, fice database ere,
te 0 0.

2.4 Un09 of he mathematica! model

"This mucel fd thee primey application in sysom analysis and.
design, Fist, it ean be wie ip make the translation which allows
‘peter cuff call objectives wo deiermine reliably objectives and
Inaintenance polcim, for ewitching and transmission systems Relis-

CUTOFF CALLS AND RELIABILITY. 1685
bility objecives should not be viewed as ends in themaclves, but only
sn means by which objectives for those aspects of customer eervice
‘hat are affected by reliably probleme can be met. Sond, thoy have
value as predictive tools. Symeem designers can use the probabilcy of
Cato ata figure of marit for hypothetical symm designs, architoc-
‘ures, and rellailicy charactevctien Systema that have not vet been
‘constructed cam be compared for this aspect of service quality, and
this compari can hea factor in deciding among competing deigns,
for example, Is third major uee ia to provide a framevore within
‘which to perform statistical tests, basod on observed eutoff call rates,
Vo sue whethur objectives ar being mt, Tw ayatetns where cutol alls
are not meesured, the models erable inferences to be made about the
cutoff all ace bused on other kinds of data, such ws relly revords
ff equipment fallares and malfunctions. Since cutoff eal are often
Aiticul or expenrve to meaaure in a given system, these Techniques
provide another, parhapemore attractive, means of understanding this
fmportane service problem,

In this sation, we discus che structs of the machernatical mada
for cucoff calls and alist of telephone equipment. It atara with an
cutlinelike guide to the sequence of results which male up the
‘mathematical model. Aran ald to seeing where the detail fic into cho
(vera cheme, this guide can ho refered to while roading the remain.
der of the paper. A quouing model with server fallures is covered, aa is
the organization ofthe servera and failure modes, Physical interpre
(stion given onl some probable insights are ded to help clarity
the ideas, Finally, the probability Gu call that bas been aecepted by
the eystom wll be cutoff is computed.

3.4 Outing of rests

‘31.1 Relation of probably of cut to equipment relabiity

"The fast important result obtained ia in Section 3.6, where the
probability chet a cll tha has been accepted by che stem wil be ext
fis computed. ‘This probability ean be thought of aaa figure of merit
for the ayetem in question, and can be computed under weak aaumsp
tione about the arrival process the holding cies, andthe incerfaihre
times. However, the probabilcy of cutoff, by itl, ia not enough to
fie « good understanding of how a system will behave with reqpect to
futting off ells. In particular, there ara two important questions on
‘which lowing the probability of cucof alone sheds no light Past,
does the observed cutoff call rae have any relation tothe probability
of cutoff? Second, wht isthe structure of the stochastic process which

1886. THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1981
counts the number of calls ea off in a cime interval? How much
‘ariability ean be expected in such a count, for example?

1921.2 Mossuremonts and concistat estimation ofthe probably of
utah

Section TV is dees tan sxloration of these questions fora mare
socialized system, he M/M/c/e queve with server flues. In aver
to the firs question, Corllvie 5 and 6 show that the oboerved culoft
fall vate converges tothe probebility of cutoff as given by og. (8). This
‘means thal, inthis ease, measurements can be relied upon to consis
tently estimate the prabability of euco, which mey be euntolled by
ie objective, Als, when a prediction about the cutoff probability in
fee ester i ae, ieean reasonably he expected chat the cuca eal
fate shown by the system in operation will approach the predicued
‘value ubjec, of course, to the quality ofthe inputs tothe prediction.

2.1.3 Axymptoticcistnbution of the number of cutot cele

Tm answer to the second question, Theorems 7 and 9 show thal the
umber of eutoff calle ig, Chen auitebly normalized, asymptotically
‘normally distcted. The azymptatic variance ofthe cumbor of cutoft
‘alla [Theorem Hy) togecher wich the agvmptatic normality, sugges
the varsity to be expected inthe observed (normalized) number of
‘lof alle stunt BS percent of observation fll within one standard
‘eviation of the mean, ete. Finally, the waympluie didebinn af the
‘nutaber of eutot ella cold he ted the bari for wali Lel
for determining whecher the objective i being me, although this ix
‘ot weeomplished in his paper.

3.2 Mathematical description of cut call model

‘The equipment will be modeled as a coerver queuing ayetem, Calle
(requests for service) arrive at the system at mes 7,75 ==» Denore
by nthe time ther ehe nth arrival entors servo, I this ea blocking
fjavem and all servers aro occupied af time Ty the ath areal never
fevers service, and for laine convenienee, rl be taken to be 29m
this cane. Throughout Section IIL the rival process may ba any
arbicnny pint process. Exch call haa asociaced with fa (nonnegative)
Tolding time that ic wiahes co apend using the resources of che system,
Tis asuined that a single call necupien only a single server in the
syscem during ieaentge holding time (Uhie will be important lar in
‘Gscusson of the organization ofthe flare ranes)."The holier
fare denoted by Y, Yo ++», and are caken ta be mutually independent
nnd ienteally discrhuted, and independent of the arrival process

'So far, we have just derenbed an ordinary quovine model. The
tudvonal festure Unt dating the madlstnciding eqs

(CUTOFF CALLS AND RELIABILITY 1867
failure i that the servers may be unralisble, That is, at certain
(random) tins oll the servers, or certain groups uf eervers may cease
serving the esters at their postions, and the affected customers
‘el be forced co depare prematurely from the system at these times.
[Adopting the naroral physical terminology for the mathematical
tote, these customers wll bo anid to huve been “eut off” Suppose
‘hae there ate m different Eallure modes in The system. That ig there
fre m diffrent ways in which various groups of servers (and possibly
all servers) ean fail in euch a way aslo cause culos at che instant the
failure begine. Any particular vorver way be affecced hy many failure
modes, and many diferent configurations of failed servers may be
inched in a single fulure mode. For example, euppose a ewivehing
system having 1200 terminations (ies and trunks) ie ade up of ten
identical unis, wach serving 120 uerminations Thon this system has a
{allure mode a 120 servers (lenainations)—this would not be ented
fs len separa failore modes all theee anit hud Ube same fire
harncteristies, With each failure mode, associate w ronewal process
Tiling the Unset which flares ofthis type orn. These me procees
will be calle “fnitare proceates” Lat 2° be the distribution of the
veal times for the ith process, and lel. be Une reciprocal of

ive bebween renewals, = ff x dF*(2) Lac the epochs
‘he (th flare process be deol hy Si, Sb +++ Tee assumed that
‘these fale process are mutually independent and independent of
(he arvivale and holding-ine proceste. The later indopendence as-
sumption i rentonable when the arrival have no prior knowledge
four the sage of the system wl the Hime of arial

‘Algo associated arith the sth failure mode is a number p; bovwoon
aero and one, The qealilyp, vepresencs che probability that weal in
the welem will be eat off when & failure of type i cecurs, and is called
the cutoff impact of failure mode "The severity of » Ture of type T
Wieleated by py Ip, = £ehen the ith eure rade ian entire system
Tale and, oth probability one, all calls In service ave cut off when.
sch a fllure occurs, the other hand, ie slase co aero chen this
describes a minor failure, und fewer calls wil he cut of when such &
Tnilane nocur. Wa wl take pO for every ‘since a failure mode with
cetaf impact zero can be ignored,

2.3 Conespondence mn physieal situation

Lmagine 4 onl osing the resources of eame telephone sytem (Cor
detinitenes, say a swilehing sce), in either the sctup phaso or che
tonverstion (sable) phase, Many elements of the syetom are used t0
brovide and rosin che conversation path Ubal iy the sleetrieal
‘connection from one sce (incoming or originating) of the system lo
‘the other (outgoing or terninaing). Failure of ame ofthese elements

1988 THE BELL SYSTEM TECHNICAL JOURNAL, OGTOBER 1081
may ease thecal tobe dropped from the cystem without an on-hook.
bycither nary. Inthe quouing model itis not these clement thal are
‘thought of aa the servers, Rather, » single all Es thought of as
fceupying « single server, such as m pair of (erninations or 2 path
through a system, which may be subject to being disabled by the
failure of some of these elements, From thi point of view, any
partiular server may’be affected hy eeveralfllure modes,

2.4 Probable interpretation

efor
thas been accepted by the aystem wal be ext off the following
Dillte heuristics are offered ab wn wil Ln clarifying the idea of the

model

"The event that a callin tho system is eut off ean be conceptualised
asa realization ofa competition proceas, Supp wcll having baling
time ¥ enters the system at time z At the entrance tine fm elucks
fare set runing, with the th clocks running time having the distri
bution ofthe excess lifetime of the time between failures for te fth
feure process at time Ifthe holding time ¥ expines before any of
the clocks run down, ne flues ovcur and, hemos, no culo can occur
Ione of the clocks rons down fir. uty the jth ene), a biased coin
(Pheads} = pis coseed. I the coin cots up heads, the call is eut
Of ued the experiment sys for tis all, T Ucn comme up
‘tho cal is not cut off, an the experiment continues, with the Je clo.
‘nn running according co the distribution J”. Por this eal, che exper.
Jmentstape either when ics beon out afar hen e departs normally
‘om the avec

"The computation, which is performed in the next section, followe
this dascription by fist detormining the probubiliy of no cutoff and
‘then subtracting from one

2.5 Probably of eulott
‘With this sceion, we begin following the outline of Section 3.1.The
sequence of resuls and thee prota is imply # mathematical anal
‘Gon of the deceription given in Section 24. Lemma 1, while of inde
pendent interest, is used here only in establishing the main result of
this section, which is Theorem 2
Lemma 1: Let {N(0): ¢ = 0} be a renewal counting process with
interrenewal time distribution F. Then or y= 0 and h = 1, the
‘probability that there are k renewals in the interoal 1, yi gven
by

[CUTOFF CALLS AND RELIABILITY 1068
here

sata yi= | CRs 9

x) - Fut y= adeis), 2)

sith Fe the bold convoltion of F with ite, Fs equals V, the
Standard right continu nat step function eh jump atthe orn
dent My the augmented renewal funcion for the process, For k= 0,
Ite probabitity that thee are no renewals tn this rer i given
JU Fee y~ his

Theorem 2: bot Bf be he augmented renewal faction for the dee
tive distribution (1 —

Hite) = B02)", ®
ot

‘Then the probity thet «call entering the system at time Cis cu off
is given by

Lf [fifleesanlans,

Inthe lit ast approaches infinity, this becomes

sini],

1 ehe arival proces i independent of tho romaining queuing and
(allure processes, che probability thatthe nth call wll be eu of iver
‘hac it enter the aystem, can bo computed by integrating 0, (5)
fgainae the ditrbution of wi. Tn eat all the failure processes are
ationary Poisson processes, ihe probability of cutoff i eonstane and
does not depend on the entrance time of che call

Corollary 3: Suppose Fiz) = 1—«™ for im I, +++. Then every
call in the system has probity of euloff wine yy

-[oel-Erm)ann

1, in addition, the eall-holding:time distribution is exponential,
Hy) = Le the probability of cutoff reduces to

14870 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1081