The analysis of call center model in case of overload Текст научной статьи по специальности «Математика»

THE ANALYSIS OF CALL CENTER MODEL IN CASE OF OVERLOAD

Sergey N. Stepanov,

MTUCI, Moscow, Russia, stpnvsrg@gmail.com

Maxim O. Shishkin,

MTUCI, Moscow, Russia, mackschischkinl@yandex.ru Georgiy. K. Sosnovikov,

MTUCI, Moscow, Russia, sosnovikov.georgy@yandex.ru

Mikhail S. Stepanov,

MTUCI, Moscow, Russia, mihstep@yandex.ru

Leonid A. Vorobeychikov,

MTUCI, Moscow, Russia, voroleonid@yandex.ru

Hanna M. Zhurko,

MTUCI, Moscow, Russia, hazhurko@gmail.com

DOI 10.24411/2072-8735-2018-10318

Keywords: call center, system of state equations, performance evaluation, multi-skilled routing, repeated attempts.

The mathematical model of call center functioning is constructed and analyzed. In the model multi-skilled routing based on usage of one group of operators for simple requests and two groups of experts (consultants) handling more advanced topics is taken into account. Call center model is considered in case of overload. It means that large portion of coming requests are repeated requests caused by insufficient amount of operators, consultants or waiting positions. In case of blocking or unsuccessful waiting time a subscriber with some probability can repeat a call. Primary and repeated requests for servicing are coming after exponentially distributed time intervals. Customer service time maximally consist of three phases: listening the recorded message from the IVR, receiving the information of general character from an operator, and getting specialized information from a consultant of the chosen group. It is supposed that all random variables used for the model description have exponential distribution and are independent from each other. Markov process that describes model functioning is constructed. The definitions of main performance measures are formulated through values of probabilities of model's stationary states. Algorithm of performance measures estimation is suggested based on solving the system of state equations by Gauss-Zeidel iterative algorithm. Expressions that relates introduced performance measures in form of conservation laws are derived. It is shown how to use found relations for indirect measurement of the intensity of primary requests and other characteristics of call center functioning. Numerical examples are presented.

Information about authors:

Sergey N. Stepanov, professor, doctor of science, MTUCI, head of the chair of communication networks and commutation systems, Moscow, Russia Maxim O. Shishkin, graduate student, MTUCI, the chair of of multimedia networks and communication services, Moscow, Russia Georgiy K. Sosnovikov, docent, Cand. Tech. Sciences, MTUCI, the chair of informatics, Moscow, Russia

Mikhail S. Stepanov, docent, Cand. Tech. Sciences, MTUCI, the chair of communication networks and commutation systems, Moscow, Russia

Leonid A. Vorobeychikov, docent, Cand. Tech. Sciences, MTUCI, the chair of informatics, Moscow, Russia

Hanna M. Zhurko, PhD student, MTUCI, the chair of communication networks and commutation systems, Moscow, Russia

Для цитирования:

Степанов С.Н., Шишкин М.О., Сосновиков Г.К., Степанов М.С., Воробейчиков Л.А., Журко А.М. Анализ модели call-центра при перегрузке // T-Comm: Телекоммуникации и транспорт. 2019. Том 13. №10. С. 68-76.

For citation:

Stepanov S.N., Shishkin M.O., Sosnovikov G.K., Stepanov M.S., Vorobeychikov L.A., Zhurko H.M. (2019). The analysis of call center model in case of overload. T-Comm, vol. 13, no.10, pр. 68-76.

1. Introduction

Call centers are used to organize the communication channels between customers and serving companies and between citizens and public organizations. Such companies are operating in the field of transportation, communications, sales and so on. We can also include in this list analytical centers that process the information of safety character obtained from different kinds of smart deices belonging to Internet of Thing.

it is well known that the most part of operational costs of a call center is the salary of operators. The optimal planning of servicing staffs a crucial problem for administration of call center. It cannot be solved without constructing and analyzing a mathematical model with taking into account the main features of call center functioning. Main among them are; the possibility of waiting, the restriction on maximum allowed time of waiting, the division of operators with respect to their skill level; the possibility of repetition of a call after blocking or unsuccessful waiting. Last feature is very important for analyzing of call center functioning in case of overload when the signifcant part of arriving requests arc retrials. The functional model of call ccnter is shown on Fig, 1.

tall (itJlt • l rri <"h1111j tic in rnc "1 l(w im

4 mine uf pi ;it.i> .!

* (K iliimnim of repealing call L

4 icnjng ay [hr i £ :r. I ur tun ml I onl

* HitEbnf i'T Kamfl Ijl u if IffIll

2

№

IVR «in «nt a ™||cr 'cq«W »illi

LiffL'd prnliatiililv

(ilfilip uf UKUHlli

>hli tu[HUlU»t, T]

Fig, 1. Functional model of call center

in tcletraffic papers theoretical analysis has been done for some of the mentioned above problems [1-11], including the study of retrials [6-11], the study of possibility of waiting and the investigation of dependence of the requests servicing on the skill level of operators [1-5]. In this paper, all mentioned factors are considered together.

In Section 2, the model functioning is described. !n Section 3 we give the formal definitions for the model's main performance measures. !n Section 4 the system of state equations is constructed. Next section is devoted to indirect measurement of intensity of primary calls and other characteristics that are dificult to measure because of necessity to distinguish primary and repeated calls. In Section 6 Gauss Seidel iterative algorithm for estimation of model stationary probabilities is presented. Numerical results is considered in Section 7. Last section is devoted to conclusions.

2. Mathematical model of call center functioning

Call center model consist of one group of operators and two group of consultants. Let us denote by u the total number of operators and by l\ and u, the total number of consultants in

the first and second groups correspondingly. Requests for getting information service enter the call center by access lines and further are served by IVR (Interactive Voice Response) and if this is

required by operators or by consultants from selected group. In case of blocking a request occupies waiting position. Let us denote by vv the number of waiting positions for operators, and

by R-'j and W't the total number of waiting positions for the first

and second groups of consultants correspondingly.

Call center model is considered in case of overload. It means that large portion of coming requests are repeated requests caused by insufficient amount of operators, consultants or waiting positions. Let us suppose that in the model the time of waiting is restricted by some random variable having exponential distribution with parameter a if customer waits free operator and parameters a, and cr1 if customer waits free consultant

from the first and second groups correspondingly. Primary requests for servicing arrive according to a Poisson process with intensity A. In case of blocking or unsuccessful waiting a customer with probability H repeats the request for servicing after random lime having exponential distribution with parameter //

and with supplement probability 1 -H the customer stops his attempts to get the service. Repeated attempt is served in the same way as a primary call.

Customer service time maximally consist of three phases: listening the recorded message from the IVR, receiving the information of general character from an operator, and getting specialized information from a consultant of the chosen group. Let us suppose that continuation of the last two phases has exponential distribution with parameters /.i and uk correspondingly,

k — 1,2 . The selection of phases depends on the type of the request: primary or repeated. With probability q for primary

requests and with probability qr for repeated request a customer after getting sen'ice ftom IVR with probability ak tries to get service at k th group of consultant, k = 1,2 and with probability a = \—at — el-, tries to get service from operators. With additional probability 1 —q for primary requests and with probability 1 —qr for retrials a customer leaves the system unserved. After finishing of operator service a customer with probability cck tries to continue service at k th group of consultant, k =1,2 and with probability 1 — c a customer leaves the system unserved. Model's main parameters and the process of requests coming and serving are shown on Fig. 2,

Let us denote the state of the system by vector (j,/,/,,/,) where j is the number of subscribers in state of repetition, 1 is the number of occupied operators and waiting positions, i, is the number of occupied consultants and waiting positions in the first group of consultants and i2 is the number of occupied consultants and waiting positions in the second group of consultants and waiting positions. The components of (7,/,/,,/-,) are taking values / = 0.1....,; / = 0,l....,v + w; ^=(>,1...../¡=0,1.....u+w-

All such vectors (j,i,irL) make up the space of states S for the model under consideration.

'-q. t + i-ip

Primary calls

A

I (a)—

[Qt* i-,

Finishing servicing < t-H

Choie« ol co niella ni group aflftr IVR

©

: s

Waung posioôTo Vjy c2

ra O,

©

clj ui,. . ^

WattJngpiocM

Choie« of consultant group *

after operator»

1-M

0S

©I

J

««[Ingpaslliû™ \ <y

H

1-«

1-H

' ■' 1J Hfvtdng

Fig. 2. The structure of call centcr mathematical model

The mode I functioning is described by Markov process r{t) = U{f),i(t)Ji{i),i2(t)),

where j(t) is the number of customers that at time / are in the state of repetition, /(/) is the number of operators and wailing positions occupied at time f: /,(/) is the number of consultants and waiting positions in the first group of consultants occupied at time t and /„(/) ts the number of consultants and waiting positions in the second group of consultants occupied at time t. The Markov character of r(!) is due to the fact that all random variables used for description the model functioning have exponential distribution and independent from each other. Lei us denote by p(j,/,/,,/,) the probability of stationary state

p(j, /, i'| ,/,)eS of the model under consideration.

In order to guarantee the existence of stationary regime it is sufficient to require that H < I and u > 0 . If these inequalities are valid than in ease of overload a customer leaves the system with probability tending to one after some time having bounded average. The values p(, /,) have interpretation as the

portion of lime the model slays in the stale {/,). This property allows to define main performance measures of the analyzed model through values of ( /,/, /".,i2 ) and model's input

parameters. The values of pM,i,i] J2 ) can be found from the solution of linear system of slale equations (1) that relates

p(jJJvi2)-

3. Main performance measures

Lei us suppose lhat we know values of p( /,/,(,,/,)£ S.

This fact allows us to define main performance measures of the analyzed call center model. !l can be done by using the interpretation of p( j,ij\j\ > as a portion of time the model slays in

the slate (_/,/,/,,/,)and technical meaning of the characteristic considered.

Mean number Mr of customers repeating a request for servicing becuuse of Mocking or unsuccessful waiting is defined as follows

Mr = X pijJJ^U)].

Ujj.it.i^eSUM))

Mean number ;V/. of occupied operators can be found Itoiti the expression

U'i = £ p(J, i, i,, i2 V + Z A H , ¡2

|(/,Mt.h)eS|l<u| |l l,ijt,h l&SlrM.'l

Mean number Mw of occupied wailing positions for subscribers waiting service at operator has the following formal definition

= Z pU.iJ^i.Jii-u).

|(Ml„(j)s№ifl}

Mean numbers of occupied consultants Mn and Mi , of

the llrsi and the second groups correspondingly can be found from formulas

M,.\= S /HÀMp';V| + Z PUJ^h^iii

il/.M|.'j )&SV|iii ! ÎU-WJ] I

Intensity of requests lh blocked because all available operators and watting positions are occupied can be found from relation

h = Z )a.

|( iJJ, J, Mïlr-u-Mi'l

Intensities of requests !h, and 1 h -, blocked because all

available consultants and waiting positions of the first and ihe second groups correspondingly are occupied can be found from relations

hi. = Z P(J*i>it,'2)Uqp+jvqr)a, +

+ Z pUJ^LMpcc^

It/.'.<1.11 letter>.:,-ny, \

+ Z p(j,i,i„i2ppccy,

II M.f| u4 lEÎ|rÏLlii^U, -lï| I

/„, ■=> X PU*< < h - k HM-P+m, + + Z p{j,i,i,J1)ipccz +

!<]JJ\ ,ii lï-Sl/ÏL».;, »il1,1

+ Z p{.i,i,i{,uy>pcc\.

!i ; ■.i. K jji '' < Il ■ j

Intensity of requests Ih g blocked when entering a cal I-center

= Z P(J> h, h + Jvlh )« + + Z pu* '' ' h )( typ + jvqr )«, +

+ Z /'(-/' h i\ > )( + Pq, ■

IU.i./|.ij )aSlii=<ijj+WJ J

Intensity of requests /r coming to get service from operators can be delined as follows

70

If = + X PO\ h h - fi \M,a-

ÍUrfU, ,iiteS)

Intensity of requests /f coming to get service from operators or consultants is defined as follows

= Mr + X V'h'"1 .•

Intensities of requests /(, and /, , coming to get service

from consultant of the first and the second groups correspondingly can be defined as follows

+ X PU'iA'hVWA +

lt/.¡,í|,lj|&5t

+ V p(j, i, /',, /, Vficq + X />0"« 'j 'i •J^&Cj i

11 ;Jj, IES(Í(J i II7./.Í, .li )eS|i>(i|

A j = + X '< 'i-'- +

leff!

+ X PÜ-1 V h V/'«* + X ^ i' '« - 4 K-V«'^ -

tt/ifJi ^ Ji^tKW f I (/.I".fag le fUfkO I

Portion of time 71, when all operators and waiting positions are occupied is delined as follows

X PUiM^h.%

lkJJ.1,.(. IÍS|J=II+ii-|

Portions of time 7t , and 7Z"f, when all consultants and waiting positions of the first and the second groups correspondingly are occupied can be defined as follows

Kj = X pU'i'hWi

ll.íjleílíl =11, -ii, I

= X PUlWnh)-

H TI'. «u. ¡

Portion 7T of lost primary and repeated requests coming to get service from operators can be defined as follows

71 =■

h + M„er

A... + M...Q-I

t, =

/»-/ft

7m - 7/u

; r, = ■

=

Mean time of being in the system on waiting or servicing

for requests coming to get service at call center can be defined by the following expression

M, -I- M„ + A-/,. + MH,, + A/,: + A/hll

A,S

Mean number MH., of requests waiting the beginning of service is defined as follows

Mean number MI of requests sen icing at call center can be

defined by the following expression A/,, = A-/, + A/,, +

Portion 7Tr of calls rejected to establish connection with operators or consultants

(Ih + M№<r + !ht + Mu + Ihl + Mn2a: )i\-H)

ft. = -

Mean number M of retrials per one primary call

/1

Mean number Q of rejections per one completed sen icing from operators or consultants

h + + IiA + Af^er, + /u _+

Q=-

A 1,fH\-c)+ M^ii+M,^

Portions K ! and 7T_ -, of lost primary and repeated requests

coming to get s en ice from consultants of the first and the second

groups correspondingly arc defined in the following way

+

'i, I h.2

Mean time tw of being in the system on waiting or servicing

for requests coming to get senice from operators can be written in the following way

M, + A-/,.,

Mean limes tu , and tw , of being in the system on waiting

or servicing for requests coming to get senice from consultants of the first and the second groups correspondingly are defined in the following way

M., + M ,

. __r. I M'hP , t _

'll.l '

Kl ~ 11),2

Ratio of lost primary and repeated requests 71. f coming to get service at call center is defined as follows

4. System of state equations

The model performance measures arc expressed through values of p(j, /| ,i2) that can be found from the solution of the

system of state equations. System of state equations is obtained by standard procedure by equating the intensity of leaving arbitrary model state (/,/, it f i2) S to the intensity of entering the

state { /,/,/,,(,), In order to solve the system by standard algorithms of linear algebra it is necessarily to limit the number of unknowns. For doing so it is enough to limit the number of customers repeating the attempt by some integer jm . The value

jm can be found through numerical experiments. When calculating the performance measures it will be supposed further that value of J varies in interval j = 0,1 ,—,jm ■ We call Ihis model as truncated.

The behavior of truncated model is the same as initial model in all models states when j < jm. If j = Jm a subscriber slops

attempts to get senice with probability equals to one. As result for truncated model we get the following system of linear equations that is expressed by one relation by using indicator function /(■) that is defined in the following way

¡1, if condition formulated in brackets is fulfilled.

0, if this condition isn't fulfilled.

Let us denote by P(j,i,,U ) the unnormalized values of p(JJj\,i2) and construct coefficient L for p(J,i,,i2) in

the left part of system of state equations.

Acceptance of a coming primary call gives a term /, = /{/ < v + w)Aqa + /</, <t\ + vv, )Aqpa{ +

+lii2<u2 +

Acccptancc of a coming repeated call gives a term l2.-I(i<V + \v)jvqra + I{ i, < u, + vt'( )jvqra^ +

+/(/, <v2 + w2)jvqra2.

Blocking and subsequent repetition gives a term

/, = /(/< u + W, j < 4 UHqra + /(/, <(J,+ H',. j < rm) X

xAHqpat + I(i2<u2 + wjTj < rm)AHqra2.

Blocking and subsequent stopping of repetition gives a term

/, = /(/ = u + w)v( I - H )qra +1 (/, = + \\\) x

xv(l - H )qral fl(i2~u2 + vv, )v(l- // k/ro,. Serving by 1VR of repeated attempt gives a term

4 =Mi-qr)-

Serv ing by operator or consultant gives a term

/,, = /(f < u)ip + /(/> u)up + /(/, < U,)/,//, + /</,> o, Uyz, +

+ /(/,< u, )/,//-, +1 ii2>u2 )u2p2. Leaving of the waiting positions gives a term

l7 = /(/> u){i~u)cr + /(f, > fj, )(/,-Ut )<7, + /{/, >U, >(f: -U, )cT,.

As result we have L=lt+l2 +/3 +/4 + /- + /() +/7. Let we pass to the construction of R the right part system of state equations for state (j,/, /,,i2).

Acceptance of a coming primary call gives a term

i] = /(/>0)p(j,i-li,,i2)Aqpa +1 </| >0)p(j,i,il-l,i2)Aqpa] +

+/(/, > 0) /?(/, /■, -1 .

Acceptance of a coming repeated call gives a term

r, = I(i>Q,J<rm)/?(./ + U-1,/,, A )(/" + l)i^r£i +

+■/(/, > 0, y < rm )/j(7> 1./,/, -I, i2 )( j + i)vqra} + +/(/,> 0, j < rm) p( j +1. /. i,, /, - \)(7 +1)vq, cu. Blocking and subsequent repetition gives a term r2 = I (/ = v + v) pij -1, /, /,, j, )XHqpa +

+/(f, = ¿j, + vi', )p{j -1, i, /,, i2 )AHqpcf, + +/(/, =u2+ w2) p(j -1,/,/,, i2 )A Hqpa2. Blocking and subsequent stopping of repetition gives a term = vv, j < rm )p(j + \,ljl, i2)(j +1)v( 1 - H )qra +

+/(/,= u, + tv,, y < rjp( 7 + L/,(,,/,)(j +1 )v( ] - H )qr a, +

+f(i2=u2+w2, j < rm)p(j + I,/,i[,i2 M J +1)v< 1 - H)q„a2.

Serving by IVR of repeated attempt gives a term

>\ = Hj< rm)(p{ j + l i, 4, i2)(j +1) v( 1 - q r)).

Serving by operator gives a term

r(i =/(/ + !< v) p(j, /+!,/,, i2)(i +1 )p{\-c) +

+}(i + \<u + wj' + \>u)pi j, i + 1, /,, i2 )vp( 1-£'V Transition to the first group gives a term

r, = i(i+1 <tJ,(, > 0)p(jj+],i, -1./, )(/+ l')c//c, + + /(/+ 1 <Ujt +11', ,j <rjp(./,/ + !,/,, /;)(/+ I -//) +

+/fi+1 £t>, i, - U, +11',, j - rm ) p(j\ i +1, i,, i2 )( f + Uf/zc, + + / (i+ 1 < Lh /', = V, + W, ,J>0)p{jJ+\, /( • i; )(i +1 )c//f, // +

+ /(i +1 <u + iv.f +1 >D,j, ><))/>{,/,( + Li, -)vcpc( + +/(f + l < u + u',1 + 1 >£>,/, = tJ, + ii',,/</*m)/j( /,/ + !,/,,

+ 1 )rcy;r,(l - H ) + /(/+ 1 < y + it'. / + 1 > v. f, = u, + iv,ij = = rn)p{jj+ I. /,, i2 )( ' + I h'Cfic, + ¡(¡+\<u+ I\\i + \> u, v'| = = U| + H',. j > 0 )p{ j,i+1, , L )( i +1 JvCjWCj H, Transition to the second group gives a term =I(i+\<vj\ > 0)p(J, i +1, /,, -1 )(( i+1 )c/ic2 +

+/(Î+1 < = U + >i':../ < rm )pij, / + !,(,,?; X ; +1 )c-/it', ( 1 - //)+ +/(/ +1 < u, i2 - u, + U':, j=r„ )/?(./■ f +1 , h XJ ■+1 + +/( f +1 < u, î, = u, + w,, / > 0)p{j, i+\j], i- H' + l )cpcJ! + +/(/+1 <u+ »',/ + ] >u,L_ >0)p(j,î +1,i,-1 + +/(i + l<u+it',/ + l >u,/: = l», +H':,/</;,)x x pij, i+1, l'y, 1 +Ï )vcpc2 (1 -//) +

l<u+w,/ + l >u,u +h'„7 = î;,1)@ x/Hlj,i+li],i2)(i + l)K'pc1 + +/(i+l<u+wj + \>u,i2 = u^ + w2,j>0)x

*pij>i+\Jx,L Xf +1 yxtifytl. Ser\ring by consultant gives a term r9 = /(i, + 1 < (J, )/£/,/,/; +1,/,){/, +1)//, + + /((, + 1 < Ui)p(j,iJl,/2 + I)(ïj + 1 )//; + +/(/, + !<[>,+ vv, ,j] +1> ¡Jj )p{jj, +1, )ty/, + +/ ( i2 +1 < u2 + »',, +1 > u2 ) p( j, i, /,, +1 )u2 p2. Leaving of the waiting positions out of call center or to repetition gives a term

/]„ = /(/ +1 < u + it; i + I > u, j < rm )p(J, i+\,i], /, )(/-i-1 — v) *

xtr(l-/y) + /(i' + l < u + m\i + 1 > u, j - rn )p( j,i +1.,L )x x(/ + I - v)<r + /(/'+1 < u + VI', / + I > u, j > 0)p(j -lji+I,/(. i2)x x(/+ I -v)aH + /((, + 1 < u, + if,./, + I >ut, j <rm)x x pi j* i, /',+ Î./,)(;',+1 - v, )cr,<l - H ) + +/(/, +1 < u, + it',, i, +1 > (j, .j = rm )p(j, i, i, +1, i, )(/, 11 - v, )* xtr, +/(/, + I < u, + »',+ 1 > , j > 0)p( j -1,/,/, +l,ij)x x(/t +1—V, )a[H + !(i} + ] <u, + w2,i, +1 > Lh, j < im)x X7)(7,Î,Î,,4 + 1 ){i2 + I - v, )(T:(! - H ) +

+/(/, +1 < u2 + vf2+ 1 >(A. j = rm )p{ j,i,/,.(,+!)(i2 + 1 - v, ) x xcr, + /(ij +1 < u, + w1,i2 +1 > , / > 0 )p(j -1,(, (",, i2 +1 )x + I - r, )(T,//.

As result we have = + +/]„■

System of stale equations connected by the following:

Pij,Ul,i2)'L = R. (])

Found values of P( /,/./,./, ) should be normalized for eal-

culating model's perfonnance measures. Introduced performance measures can be expressed through few of them by means of

72

relations that we call conservation laws. They will he derived in ihe next section.

5. indirect measurements of performance measures

Model performance measures are related by Conservation laws that can he obtained after multiplication of the system of

stale equation (I) by /, i, Z(, L and consequent summation. For simplicity we continue to suppose that jm — so . After collecting the terms we get four relations according to the number of components in the vector of state (/, ,/■,):

Mrv = (4 + Afjt + /„.i + A^ jii, + Ih 2 + MntZ<J2 )H:

I, = Ih + M„cr + MLtl;

A., ='/,,) + ^'.i^i + AW

Obtained relations have very simple explanation. The first relation says that intensity of repeated requests equals to the intensity of events that leads to the repetitions, in the mode! we differ two such events: blocking and unsuccessful waiting. The second relation says that intensity of requests coming to operators equals intensity of blocked, leaved and served requests. The same is true for consultants. The last two relations say that intensity of requests coming to the first and the second group of consultants correspondingly equal the intensity of blocked, leaved and served requests.

The introduced performance measures and model's input parameters can he expressed through some of them by means of found relations (2). Especially valuable arc expressions that allow us to find characteristics and parameters that are difficult to measure by direct methods because in doing so it is necessary to divide the total flow of coming requests into parts corresponding to primary and repeated components. It is hard to do for big eali centers because we should keep the history of contacts of each customer with call center. However we can solve this problem indirectly by relations derived from conservation laws (2). Let us show this on the example of indirect measurement of

By using standard measurement equipment we can easily estimate the value of /( the total intensity of primary and repeated requests entering call center to get servicing from operators or consultants and the value of 7T ( the ratio of lost primary and repeated requests coming to get service at call center from operators or consultants. In measuring it and JT._ ( we do not have to distinguish primary and repeated requests. By multiplying the first relation (2) on t/r and adding to the lefi- and right-hand

sides of this relation the Value of Xqr and using the definition of ncS, we get /f = I,7rctHqr + Xqp.

From here follows formula for indirect estimation of A

q' (3)

In a similar way we can derive formulas for indirect estimation of other characteristics of call center functioning that are

difficult to measure by straightforward methods. For M the formula is looking as follows

7T , 1. H

M. = CJ ' .

M

For /T. we obtain

= jrcJI,(i-H) = irejq,.(!-//)

* \-H(lr7TaJ

The expression for M can be derived in the following way

_ Mv ^ xt,,H(lr & \-Xc,Hqr*

In order to find the formula for 0 we need to make some transform of (2) It is clear that only CCt and CC2 portions of requests serving by operators enter the first and the second group of consultant for continuation of service. It means lha the following relations are true

(41

Adding last three relations of (2) and (4) and collecting the terms, it is easy to see that

/, = /, + Mua + /„ , *Mw]at + + MVJa2 +

+4, + M^O-c) + MuJut + ML1p2. t5)

Relation (5) has an obvious physical meaning: the intensity of requests coming to get service from operators or consultants equals the sum of intensities of the Hows of blocked and served requests by operators or consultants. From (5) and definition of Q follows for Indirect measurement of this characteristics

Q= /Jt + M,o-+/M + Af^,cr, + /^ + A/,ii:<r: = jr., /, - (/„ + A</„ <7 + /,,, + JM^jflj + + &) I - jt, ,'

ft. Calculation of characteristics

To find the model perfonnance measures according their definitions it is necessary to calculate values of p( /,j\ip(2) that

are related by system of linear equation (I). Most elements of the matrix of the (1) are zeros. In such a ease the efficient procedure to solve (1) consist in using Gaoss-Seidel iterative algorithm. Let us mark by L(j,i,i,,/,) the coefficient for P( jt/, i,i2) in the left part of (I) and by R{P[ /,/,/,./,)) we mark the right part of {1). The /.( /,/,/,,/, I is a function of the integer components of the state (/,/,/",,/,) e 5 . The /,/,,/,)) is a function of

probabilities of the states (/,/,/,,/-,) from which r(7) can transit by one step to ( /,i,L, L). They are gi\ en in the right part of (I).

It is not difficult to derive Gauss-Seidel recursions for calculation of the unknown probabilities P( j,ij\J,). Let us denote

by P[ l{ j\ i J,, ) ihe S th approximation for P( ) ob-

tained by Gauss-Seidel algorithm. The initial approximation can be chosen from relations P""(/Jj]J-,) - I. (/*, /,/t,/, )eS . The sequence of approximations is obtained from requrrence:

i. (, ,(:)= - , ' . . R( I, /,,/; )),

UjJ,i„i2)

Upper index (s.s + I) in (6) means using of last known approximation for considered probability. Convergence is verified by expression

X (PlstnuM,i2))

ieS

where € is taken from interval 10 ,th___10" ' " . Details of using

Gauss-Zeidel algorithm for solving system of state equations are discussed in [ 12-14].

7. Numerical analysis

Planning the number of operators. Main problem in analyzing the statistical data obtained by measurement at call centers is to separate the primary and repeated calls. In principle it is possible to differentiate calls with respect to the history of their interactions with the call center, but it leads to significant computational costs for the storage and processing of information. Because of this reason it is seldom done. Such practice may lead to significant errors in estimating the necessary number of operators and consultants. Let us show this on the example of constructed model. Let us suppose that in the process of call center functioning it was found that the number of operators is less than

necessary and cause the increasing of 7tc t the ratio of lost calls.

To simplify the problem we suppose that number of consultants in each group is sufficient to serve coming calls. A traditional procedure to find the necessary amount of operators is to sequentially increase the number of operators v and estimate /r ( for

all intermediate values of V and fixed values of other model parameters in particular the intensity of coming calls. The minimal value of v for which n , < n will be the answer of the

c,< nonv

problem. This approach significantly overvalues the desired value of v . This happens because in reality the intensity of coming calls consist of two components. One corresponds to the flow of primary calls. The intensity of this flow does not depend on the current number of operators. Other represents the fiow of repeated calls. The intensity of this flow depends on the number of operators in particular decreasing when number of operators increases. This separation of the incoming calls flow can be taken into account only with help of models with retrials (see formula (3)).

Let us consider a numerical example that illustrates the solution of the problem of estimating the number of operators with the traditional approach and an algorithm based on counting repeated calls. The analyzed model of call center has the following values of input parameters: q t = 0,5; qr — 0,9; o, =0,2;

o2 »0,1; q -a, = 0,7; /f-0,9; v=12; ./,„=20; // = 1; //,=0,75; /(,=0,5; c = 0.2; c,=0,6; c2=0.4;

a = 1; rr, = 0,75; a-, = 0,5; u~ 8; w =

o. =12;

iv, =6; u, =1; h't = 5; £ = It)1 . The values of characteristics found for jm - 20 are excepted as exact values corresponding to the case of unlimited interval of changing / . As a time

unit was chosen the mean time of serv icing a request by operator. The calculation of necessary amount ofoperators with taken into

account retrials shown on Fig. 2. We suppose that results of measurements have got the value of the total intensity of requests coming equals to 54,06 calls per time unit. Using the traditional approach, e.g. by assuming that all coming requests are primary we obtain that it necessary to use 23 operators to get the value ft less than 0,05. When we solve formulated problem with

help oT model with taken into account the retrials we obtain that it necessary to use 17 operators. The difference is quite big. The results of calculations arc presented on Fig, 3. It is clear that the error in our estimate of the necessary number ofoperators will increase with the value of repeat probability.

is n a

Number of operators, U

Fig. 3:

The calculation of necessary amount ofoperators with out taken into account retrials

The dependence of performance measures on the distribution of time interval between successive retrials. Mathematical analysis of call center functioning is usually done by using exponential assumptions concerning distribution of time interval between random events changing the model state. Among such events are: coming of primary calls, coming of repeated calls, duration of service phases, maximum allowed waiting time an so on. Exponential assumption gives the possibility of using Markov processes for describing the model functioning. This allows to find per fo nuance measures by solving the system of state equations with well elaborated algorithms of linear algebra. On of the main parameter of mathematical model is the duration and distribution of time interval between successive retrials. Let us show by simulation |15| that main performance measures weakly depend on distribution of time interval between successive retrials.

The analyzed call center model input parameters are taken as follows: A - 40; qp=Q,5; (/, =0,7; a, = 0,2; a, =0,1;

0 = 1-0,-0, =0,7; tf = 0,9; v = 5; jm =40; fi = U

//,=0.75; //,= 0,5; r = 0,2; c, =0,6; r,=0,4; cr = l;

erj = 0,75; <x, = 0,5; v - 12; w = 6; vl = 8; w, =4; u, - 6;

iv't = 3; e - 10 l\ As a time unit was chosen the mean lime of

servicing of a request by operator. In the Table the values of main performance measures are shown depending on the type of distribution and values of variance V that is varies form 0 to 10. The values of main performance measures shown at Table. Four types of distribution functions are considered: exponential M; deterministic D: uniform U; hyperexponential /7,. For all distributions the mean duration of time interval between successive retrials is the same and equals 1/5 of chosen time unit.

Table 1

The values of main performance measures found by si m u la ( ions

Perform nee measure Markov model S mi ill a lion model

M IJ U H2

V=2 V~4 V=6 V~8 V--10

1 Mi 11.585 1 1.58') 11.591 11.576 ] 1.547 11.532 11.529 11.53 11.5

2 Mti 6.9292 <1.9 JO 6.94ft 6.941 h.923 6.915 6.919 6.933 6.9?

3 Mtz 5.204 b 5.209 5.195 5.194 5.197 5.203 5.187 5.195 5.15

4 Mu 23.719 23,748 23.732 23,711 23.667 23.650 2.i.(>35 23:660 23.6

5 Mw 2.8156 2.812 2-84! 2.801 2-672 2-653 2.651 2.617 2.68

6 MH. ! 0.9286 0.9« 0.9? 5 0.929 0.920 0.930 0,928 0.920 0,94

7 0,8U5 (J.799 0.793 0.801 0.799 0.796 0.792 0.802 0.72

8 4.5557 4.556 4.569 4.531 4J9I 4.378 4.371 4.339 4.30

9 tw i.nooo 11.997 0.997 0.997 0.997 0,997 a 998 0,997 0,99

10 Vt 1.3333 1.336 l.33(. 1.335 1.333 1.334 1.334 1.333 1.38

II V* 2,0000 1.996 1.991 2.0O3 1.998 1.999 2.001 2.1106 1,93

| 1 twi 1.3343 1.333 1.3;: 1.334 1.336 1,336 1.335 1.337 1,36

13 7TC (1.3378 0.339 0.337 0.338 0.341 0.344 0.346 0.344 0.38

1-1 ^c.l 0.1866 0.190 0.187 0.1 S7 0,190 0.192 0.191 0.189 0.15

! 5 nc2 0.2404 0,236 0.334 0.237 0.240 0.238 0.243 0242 0.242

It) 0.317! 0,318 0.316 0.317 0.320 0.322 0.324 0.323 0.32

17 <1 0.464} 0.467 0462 0.464 0.471 0475 0.479 0.476 (.1.48

IK Mr 1.4204 1 434 1,422 1.425 1.489 1,395 1.432 1.321 1.32

19 M 0.1783 0,180 0,178 0.1 78 0.180 0.182 ! 1.184 0.182 0.18

The results of simulation [15] says thai the values of performance measures are weakly depends on type of the distribution of time interval between successive retrials {the relative error is round few percents). This is very important result because it allows to use Markov processes tor estimation of performance measures based on solution of system of slate equation by Gauss-Zeide! iterative algorithm.

8. Conclusion

In the paper the mathematical model of call ccnier functioning in case of overload is constructed and analyzed. In the model multi-skilled routing based on usage of one group of operators for simple requests and two groups of experts (consultants) handling more advanced topics is taken into account. In case of blocking or unsuccessful waiting time a subscriber with some probability can repeat the request. Primary and repealed requests for sen icing arc coming after exponentially distributed time intervals. It is supposed that all random variables used for the model description have exponential distribution and are independent from each other. Markov process that describes model functioning is constructed. In the framework of the proposed model the definitions of main performance measures of interest are formulated through values of probabilities of model's stationary states. Algorithm of performance measures estimation is suggested based on solving the system of stale equations by

Gauss-Zeidel iterative algorithm. Expressions that relates introduced performance measures in form of conservation laws are derived, it is shown how to use found relations for indirect measurement of the intensity of primary requests and other characteristics of call center functioning. We have considered problems of planning the number of operators with and without taken into account retrials and illustrate our solutions for die proposed problems with examples of numerical computations. Future work may include approximate methods for computing the model's performance measures and examples of using them to estimate values of structural parameters of call center.

References

1. Gans, N., Koole M., Mandelbaum A. (2003). Telephone Call-Centers: Tutorial, Review and Research Prospects, MannJ. Service Manage, no. 5, pp. 79-141.

2. Stolletz. R,, HcIIxt, S. (2004). Performance Analysis of an Inbound Call-Center with Skills-Based Routing. Hannover: Springer-Yellag.

3. Borst, S., Maudelbftum, A., Reiman Ml. (2004). Dimensioning large call ccmcrs. Operations Research. Vol. 52, no I, pp. 17-34.

4. Koole Ci., Mandelbaum A, (2002). Queucing models of call centers: An introduction. Annals of Operations Research. Vol. 113, no 4, pp. 41-59.

5. Cc/ik, MX. L'Eeuvcr P. (2008), Staffing mutt ¡skill call ccnters via linear programming and simulation. Management Science, Vol. 54. no 2, pp. 310-323.

6. Stepanov, S.N. (1983). Chislennye melody raschcia sistem s povtornymi vyzovami (Numerical Methods for Analysis of Systems with Repealed Calls), ;\auka, Moscow, {in Russian).

1. Stepanov, S.N.. Stepanov, M.S. (2014). Construction and Analysis of a Generalized Contact Center Model. Automation ami Remote Control. Vol. 75, no. 11, pp. 1936-1947.

8. Stepanov, S.N,. Stepanuv, M.S. (2016), Algorithms for Estimating Throughput Characteristics in a Generalized Call Center Model. Automation ami Remote Control. Vol, 77, no. 1. pp. 1195-1207.

9. Aguir, S., Karaesmen, F., Aksin, O.Z.. Chauvet F. (2004). The impact of retrials on call center performance. OR Spectrum. Vol. 26. no 3, pp. 353-376.

10. Stepanov S.N. (1998). Generalized model with retrials in case of CNtreme load. QueueingSystems. Vol. 27, pp. 131-151.

I!, Stepanov S.N. (1999). Markov Models with Retrials: The Calculation of Stationary Performance Measures Based on the Concept of Truncation, Mathematical and Computer Modelling, Vol, 30, pp. 207-228.

12. Barker, V.A, (19&9>, Numerical solution of sparse singular systems of equations arising from ergodie Markov Chains. Stochastic Models. Vol, 5, no 3, pp. 335-381.

13. Stepanov, S.N. (20101. Osnovy (detrainka muldservisnykh setei (fundamentals of Multiservice Networks). Eqo-Trends. Moscow, (in Russian)

14. Stepanov, S.N. (2015). Teorlya teletraffika: kontseptsu, mode I i, prilozheniya (Theory of Teletraffic: Concepts, Models, Applications). Garyachava Liniya-Telekom. Moscow, (in Russian)

15. Vorobeychikov L.A.. Sosnovikov G.K. (2019). Modelirovanie sistem massovogo obslugivania v srede tiPSS World: ucheb. posobie Modeling of queuing systems in the simulation environment GPSS World: manual. Moscow. KURSPubt (in Russian}

АНАЛИЗ МОДЕЛИ CALL-ЦЕНТРА ПРИ ПЕРЕГРУЗКЕ

Степанов Сергей Николаевич, Московский Технический Университет Связи и Информатики (МТУСИ), Москва, Россия Шишкин Максим, Московский Технический Университет Связи и Информатики (МТУСИ), Москва, Россия Сосновиков Георгий, Московский Технический Университет Связи и Информатики (МТУСИ), Москва, Россия Степанов Михаил Сергеевич, Московский Технический Университет Связи и Информатики (МТУСИ), Москва, Россия Воробейчиков Леонид, Московский Технический Университет Связи и Информатики (МТУСИ), Москва, Россия Журко Анна, Московский Технический Университет Связи и Информатики (МТУСИ), Москва, Россия

Аннотация

Построена и исследована математическая модель функционирования call-центра. В модели учтена многопрофильная маршрутизация поступающих заявок на информационное обслуживание. Рассмотрено использование одной группы операторов для ответа на простые вопросы и двух групп консультантов для ответа на более сложные запросы. Работа call-центра исследуется в случае перегрузки. Это означает, что значительная часть поступающих заявок представляет из себя повторные обращения, вызванные нехваткой операторов, консультантов или мест ожидания освобождения операторов или консультантов. Время обслуживания заявки максимально состоит из трех фаз: прослушивание сообщений IVR, ответ оператора или ответ консультанта. В случае блокировки или длительного времени ожидания абонент с некоторой вероятностью может повторить вызов. Первичные и повторные запросы на обслуживание поступают через экспоненциально распределенные промежутки времени. Предполагается, что все случайные величины, используемые для описания модели, имеют экспоненциальное распределение и не зависят друг от друга. Построен марковский процесс, описывающий функционирование модели. Введены определения основных показателей качества обслуживания поступающих запросов через значения стационарных вероятностей состояний модели. Предложен алгоритм оценки рассмотренных характеристик, основанный на составлении и решении системы уравнений равновесия с помощью итерационного алгоритма Гаусса-Зейделя. Получены выражения, связывающие введенные характеристики качества обслуживания заявок в виде законов сохранения интенсивностей поступающих и обслуженных заявок. Показано, как использовать найденные соотношения для косвенного измерения интенсивности первичных запросов и других характеристик функционирования call-центра. Приведены численные примеры.

Ключевые слова: call центр, система уравнений равновесия, оценка производительности, многопрофильная маршрутизация, повторные вызовы. Литература

1. Gans, N., Koole, M., Mandelbaum, A. Telephone Call-Centers: Tutorial, Review and Research Prospects // Manuf. Service Manage. 2003. no. 5, pp. 79-141.

2. Stolletz, R., Helber, S. Perfomance Analysis of an Inbound Call-Center with Skills-Based Routing // Hannover: Springer-Vellag. 2004.

3. Borst, S., Mandelbaum, A., Reiman M.I. Dimensioning large call centers // Operations Research. 2004, vol. 52, no 1, pp. 17-34.

4. Koole G., Mandelbaum A. Queueing models of call centers: An introduction // Annals of Operations Research. 2002, vol. 113, no 4, pp. 41-59.

5. Cezik, M.T., L'Ecuyer P. Staffing multiskill call centers via linear programming and simulation // Management Science. 2008, vol. 54, no 2, pp. 310-323.

6. Степанов С.Н. Численные методы расчета систем с повторными вызовами. М.: Наука. 1983.

7. Степанов С.Н., Степанов М.С. Построение и анализ обобщенной модели контакт-центра // Автоматика и телемеханика. 2014. № 11. С. 55-69.

8. Степанов С.Н., Степанов М.С. Алгоритмы оценки показателей пропускной способности обобщенной модели контакт-центра // Автоматика и телемеханика. 2016. № 7. С. 86-102.

9. Aguir, S., Karaesmen, F., Aksin, O.Z., Chauvet F. The impact of retrials on call center performance // OR Spectrum. 2004, vol. 26, no 3, pp. 353-376.

10. Stepanov S.N. Generalized model with retrials in case of extreme load // Queueing Systems. 1998, vol. 27, pp. 131-151.

11. Stepanov S.N. Markov Models with Retrials: The Calculation of Stationary Performance Measures Based on the Concept of Truncation // Mathematical and Computer Modelling. 1999, vol. 30, pp. 207-228.

12. Barker, V.A. Numerical solution of sparse singular systems of equations arising from ergodic Markov Chains // Stochastic Models. 1989, vol. 5, no 3, pp. 335-381.

13. Степанов С.Н. Основы телетрафика мультисервисных сетей. М.: Эко-Трендз. 2010.

14. Степанов С.Н. Теория телетрафика: концепции, модели, приложения // Серия "Теория и практика инфокоммуникаций". М.: Горячая линия -Телеком.-2015.

15. Воробейчиков Л.А., Сосновиков Г.К. Моделирование систем массового обслуживания в среде GPSSWorld: учеб. пособие. М.: КУРС, 2019.

Информация об авторах:

Степанов Сергей Николаевич, Московский Технический Университет Связи и Информатики (МТУСИ), заведущий кафедрой сети связи и системы коммутации, д.т.н., Москва, Россия

Шишкин Максим, Московский Технический Университет Связи и Информатики (МТУСИ), кафедра мультимедийных сетей и услуг связи, магистрант, Москва, Россия

Сосновиков Георгий, Московский Технический Университет Связи и Информатики (МТУСИ), доцент кафедры информатики, к.т.н., Москва, Россия Степанов Михаил Сергеевич, Московский Технический Университет Связи и Информатики (МТУСИ), доцент кафедры сетей связи и систем коммутации, к.т.н., Москва, Россия

Воробейчиков Леонид, Московский Технический Университет Связи и Информатики (МТУСИ), доцент кафедры информатики, к.т.н., Москва, Россия Журко Анна, Московский Технический Университет Связи и Информатики (МТУСИ), кафедра сети связи и системы коммутации, аспирант, Москва, Россия