Two-way communication retrial queue with unreliable server and multiple types of outgoing calls

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1. Introduction Blended call centers are an efficient and productive implement in modern companies. In addition to the usual call center features they can provide both incoming and outgoing calls. In its idle time the server initiates an outgoing call. Such behaviour allows to increase the effectiveness of the system. To study the functioning of the system with likewise particularity retrial queues are commonly used [1]-[3]. In retrial queues the customers that find the server busy repeat their request for service after a random delay. The pool of waiting customers is called an orbit. The most extensive research of this type of queueing systems is given in monographs [4], [5]. Retrial queues are usually the models of telecommunication and random access systems and their modifications allow to describe various real systems. In this paper, we consider retrial queue with two-way communication and unreliable server. Two-way communication arose as a modification of retrial queues for modeling call centers with mixed calls. The Markovian models with outgoing calls are researched by Artalejo and Phung-Duc [6], [7]. Twoway communication queues have a branch of modifications to concrete the performance such as: finite input source [8], server-orbit interaction [9], [10], call-back option [11], etc. In our case there are multiple types of outgoing calls in the system. The unreliability is explained by the need to take into account possible interruptions in the operation of the server [12], [13]. In our definition the rates of breakdowns are different and depend on the server current state. To study the model we use asymptotic analysis method under high rates of making outgoing calls condition [14]. 2. Model description We consider single server retrial queue with multiple types of outgoing calls. Input process is a stationary Poisson process with rate . Incoming calls occupy the server for an exponentially distributed time with rate 1. Calls that find the server busy join the orbit and repeat their attempt to take the server after an exponentially distributed delay with rate . In its idle time the server makes outgoing calls of type with rate and provides the service for an exponentially distributed time with parameter . For convenience, we number the types of outgoing calls from 2 to . Unsteadiness of the system is a behaviour of the server that it crashes after an exponentially distributed time of functioning. The rate of breakdowns and restorations are distinguished and depend on the server state. Figure 1 depicts the structure of the system. ♥ … ❄ ♥ ❄ ✑ ✲ , = 1, 0, 1, 2 ❄ Figure 1. Single server retrial queue with unreliable server and multiple types of outgoing calls Let () denotes the state of the server at the time ⩾ 0, ⎧ {0, if the server is idle, { {1, if an incoming call is in service, () = ⎨ {, if an outgoing call of type is in service, = 2, , { { ⎩ + 1, if the server is in recovery state. As the server breakdowns depend on its state we denote 0 is the rate of breakdowns in state 0, 1 is the rate of breakdowns in state 1 and 2 is the rate of restorations. We assume that when the outgoing call is in service there is no breakdowns as the server calls the customer itself. If the customer is in service the breakdown interrupts the service and the customer joins the orbit. Let () denotes the number of calls in the orbit at the moment ⩾ 0. It is easy to see that two-dimensional process {(), ()} is a continuous time Markov chain. Let {() = , () = } = (, ) denotes the probability distribution of the process {(), ()}, then it is the solution of Kolmogorov’s system of equations. We present the system in stationary regime ⎧ { { - ( + + ∑ + 0) 0() + ∑ () + 2+1() = 0, { =2 =1 { ⎨ - ( + 1 + 1)1() + 1( - 1) + 0() + ( + 1)0( + 1) = 0, { - ( + )() + ( - 1) + 0() = 0, = 2, , { ⎩ - ( + 2)+1() + +1( - 1) + 00() + 11( - 1) = 0. (1) ∞ Let () denotes the partial characteristic functions () = ∑ (), √ =0 = 0, + 1, where = -1. Multiplying equations of system (1) by and taking the sum over yields ⎧ { - ( + 0 + ∑ ) 0() + ′ ()+ { 0 { =2 { { { + ∑ () + 2+1() = 0, { { =1 0 { ⎨(( - 1) - 1 - 1)1() + 0() - -′ () = 0, (2) { {(( - 1) - )() + 0() = 0, = 2, , {(( - 1) - 2)+1() + 00() + 11() = 0, { { { {-′ +1 0() + ( + 1)1() + ∑ () = 0, ⎩ =2 where the last equation is an additional equation that we obtained by summing up the equations of the system. The characteristic function () of the studied process can be expressed through the partial characteristic functions as follows +1 () = ∑ (). =0 The main contribution of this research is the solution of the system (2) by using an asymptotic analysis method under high rate of outgoing calls limit condition. 3. Asymptotic Analysis of the Model under the High Rate of Making Outgoing Calls In this section, we will investigate system (2) by asymptotic analysis method under the high rate of making outgoing calls limit condition. In particular, we prove that asymptotic characteristic function of the number of incoming calls in the system corresponds to Gaussian distribution. To match the asymptotic condition we denote = in system (2) ⎧ { - ( + 0 + ∑ ) 0() + ′ ()+ { 0 { =2 { { { + ∑ () + 2+1() = 0, { { =1 0 { ⎨(( - 1) - 1 - 1)1() + 0() - -′ () = 0, (3) {(( { - 1) - )() + 0() = 0, = 2, , {(( - 1) - 2)+1() + 00() + 11() = 0, { { { {-′ +1 0() + ( + 1)1() + ∑ () = 0. ⎩ =2 then the limit condition takes the form → ∞. 1. First Order Asymptotic We denote = 1/ in the system (3), and introduce the following notations = , 0() = 0(, ), () = (, ), = 1, + 1, in order to get the following system ⎧ (, ) { - ( + 0 + ∑ ) 0(, ) + { 0 + { =2 { { { + ∑ (, ) + 2+1(, ) = 0, { =1 { {(( - 1) - 1 - 1)1(, ) + 0(, )- { ⎨ - - 0(, ) = 0, (4) { { {(( - 1) - )(, ) + 0(, ) = 0, = 2, , { {(( - 1) - 2)+1(, ) + 00(, ) + 11(, ) = 0, { { {- 0(, ) + ( + ) +1 (, ) + ∑ (, ) = 0. ⎩ 1 1 =2 Theorem 1. Suppose () is the number of incoming calls in the aforementioned system of the stationary M/M/1 retrial queue with several types of outgoing calls, then the (5) holds where lim →∞ () = 1 , (5) 2(1 + 1) 1 = - ( + ∑ ) , (6) 1 2 1 2 =2 Proof. Considering the limit as → 0 in the system (4), then we will get ⎧ { - ∑ 0() + ′() + ∑ () + () = 0, { { =2 0 =1 2 +1 { - (1 + 1)1() - ′() = 0, { 0 ⎨ - () + 0() = 0, = 2, , { { - 2+1() + 11() = 0, (7) { { ′() + ( + { 0 1)1 ⎩ +1 () + ∑ =2 () = 0. We propose to seek the solution of the system (7) in the following form () = Φ(), = 0, + 1, (8) where is the stationary probability distribution of the system states, then, dividing the equations by Φ(), we obtain the system in the following form ⎧ Φ′() { - 0 ∑ + 0 { Φ() + ∑ + 2+1 = 0, { =2 { Φ′() =1 { { - (1 + 1)1 - 0 = 0, Φ() (9) { ⎨ - + 0 = 0, = 2, , { { - 2+1 + 11 = 0, { { {0 ⎩ Φ′() Φ() +1 + ( + 1)1 + ∑ = 0. =2 As the relation Φ′() Φ() doesn’t depend on , the characteristic function Φ() can be expressed as Φ() = exp{1}, which coincides with (5). Thus, we rewrite the system ⎧ { - (∑ + 1) 0 + ∑ + 2+1 = 0, { { =2 =1 { { - (1 + 1)1 + 10 = 0, ⎨ - + 0 = 0, { { - 2+1 + 11 = 0, = 2, , (10) { +1 { { - 10 + ( + 1)1 + ∑ = 0. ⎩ =2 To obtain the values of we use second, third and fourth equations of the system with normalization condition 1 1 ⎧ = { { { 1 + 1 0, { = { 0, = 2, , { ⎨+1 = { { +1 1 2 1 = 11 2(1 + 1) 0, (11) { { ∑ = 1 0 + ∑ 0 + 11 0 = 1. ⎩ =1 1 + 1 =2 2(1 + 1) From the last equality we obtain the value of 0 -1 1(1 + 2) 0 = ( ( + + ∑ ) ) , (12) 2 1 1 =2 using which we can write the values as follows -1 ⎧ { {1 = { { 1 1 + 1 ( 1(1 + 2) 2(1 + 1) + ∑ =2 ) , -1 { 1(1 + 2) , = 2, , (13) ⎨ = { { { ( 2 (1 + 1 + ∑ ) ) =2 -1 { {+1 = 11 ( 1(1 + 2) + ∑ ) , ⎩ 2(1 + 1) 2(1 + 1) =2 The explicit value of 1 we obtain from the last equation of the system (10) using the equations (11) 1 = ( + 1)1 + +1 ∑ =2 = 2 (1 + 1) ∑ , (14) 0 12 - (1 + 2) =2 which coincides with (6). D 2. Second Order Asymptotic We introduce the following notations in the system (3) () = 1 (2)(), (15) to obtain the system of equations (16) ⎧ (2) (2)() { { - ( + 0 + 1 + ∑ ) 0 () + 0 + { =2 { { (2) (2) { + ∑ () + 2+1() = 0, { =1 { {(( - 1) - 1 - 1) (2) () + ( + ) (2) ()- { 1 1 { 0 (2) { ⎨ { (2) - - 0 () = 0, (2) (16) {(( - 1) - ) () + 0 () = 0, = 2, , { {(( - 1) - 2) (2) () + (2) () + (2) () = 0, { { (2) +1 0 0 1 1 { {- 0 () - - (2) () + ( + ) (2) ()+ { 1 0 { { 1 +1 1 (2) { + ∑ () = 0. ⎩ =2 Denoting = 1/2, and introducing the following notations in the system (16) (2) (2) = , (2) (2) (17) 0 () = 20 (, ), () = (, ), = 1, + 1, we obtain ⎧ (2) (2)(, ) { { - (2 + 02 + 1 + ∑ ) 0 (, ) + 0 + { =2 { { (2) (2) { + ∑ (, ) + 2+1(, ) = 0, { =1 { (2) (2) { {(( - 1) - 1 - 1)1 (, ) + (2 + 1)0 (, )- { { - - { (2) 0 (, ) = 0, ⎨ (2) (2) (18) {(( - 1) - ) (, ) + 0 (, ) = 0, = 2, , { (2) (2) {(( - 1) - 2)+1(, ) + 020 (, )+ { { + 1 (2) (, ) = 0, { { { (2)(, ) 1 (2) (2) {- { { { { ⎩ 0 - 1-0 (, ) + ( + 1)1 (, )+ +1 + ∑ (2)(, ) = 0. =2 Theorem 2. In the context of Theorem 1 the following equation is true lim exp { 2 () - 1 } = exp { () √ 2 } , (19) where →∞ 2 2(1 + 1) 1 2 = 2 - 2 × - 1 2 × (1 + 1 2 1(1 + 2 + 1)( + 2) + 2 2 + ∑ ) . (20) 2 (1 + 1)2 2 =2 Proof. To solve the system of equations (18) we present the functions (2) (, ) in the form (2) (, ) = Φ2(){ + } + (2), = 0, + 1, (21) here is the probability distribution of the server state obtained in Theorem 1. Substituting (21) into (18), making some simple conversions and taking (10) into account in the limit by → 0, we obtain the system (22) ⎧ ′ { Φ2() 0 - (1 + ∑ ) 0 + ∑ + 2+1 = 0, { Φ() { =2 { ′ =1 { - Φ2() + - ( + ) + = 0, { Φ2() 0 1 1 1 1 1 0 (22) { ⎨ - + 0 = 0, = 2, , { {11 + +1 - 2+1 + 11 = 0, Φ′ { +1 { { ⎩ 2() Φ2() 0 + 1 0 - 1 0 + ( + 1 )1 + ∑ =2 = 0. 2() These equations imply that the relation Φ′ () doesn’t depend on , thus Φ2 the function Φ2() is given in the following form ()2 Φ2() = exp { which coincides with (19). 2 2} , 2() We have Φ′ Φ2() = -2 and then we obtain the system ⎧ { - (1 + ∑ ) 0 + ∑ + 2+1 = 20, { { =2 =1 { { - (1 + 1)1 + 10 = -20 - 1, ⎨ - + 0 = -, = 2, , { { - 2+1 + 11 = -11 - +1, (23) { +1 { { - 10 + ( + 1)1 + ∑ = 20 - 10. ⎩ =2 The system of equations (23) is heterogeneous form of the system (10), thus we represent the values in the form of (2) = + (2), (24) and substitute (24) into (23). According to the (10) obtained in Theorem 1, the terms containing the constant are destroyed. We get system of equations ⎧ { - (1 + ∑ ) 0 + ∑ + 2+1 = 20, = (2), { { =2 =1 { { - (1 + 1)1 + 10 = -20 - 1, ⎨ - + 0 = -, = 2, , { { - 2+1 + 11 = -11 - +1, (25) { +1 { { - 10 + ( + 1)1 + ∑ = 20 - 10, ⎩ =2 From the second, third and fourth equations of the system (25) we have the following equalities 1 2 1 1 = + 1 1 0 + + 1 1 0 + + 1 1, = 0 + , = 2, , = 1 +1 2 + 1 1 2 2 1 + +1 = = 11 2(1 + 1) 0 + 21 2(1 + 1) 0 + 1 2 (1 + 1 1 + ) 1 + 2 +1. Substituting these equalities into the last equation of the system (25) we get 1( + 1) 11 (-1 + 1 + 1 + ∑ =2 + 2 (1 + 1 ) 0+ ) + ( 2( + 1) 1 + 1 + 21 2(1 + 1) - 2 + 1) 0+ + ( ( + 1) + 1 (1 + )) 1 + ∑ + 2 +1 = 0. 1 + 1 2 1 + 1 =2 2 As the coefficient at 0 is zero we can rewrite this equation to obtain the explicit expression of 2 which coincides with (20): 2(1 + 1) 1 2 = 2 - 2 - 1 × 1(1 + 2 + 1)( + 2) + 2 + 2 ∑ ) , × (1 + 1 2 2 2 2 2 (1 + 1) =2 To express the characteristic function () of the number of customers in the orbit we use the notations (15) and (17) () = exp {1 + (()2/2) 2} . Thus, in Theorem 2 we have obtained the variance 2 of a number of calls in the orbit in prelimit situation of → ∞. The asymptotic probability distribution of the process () is Gaussian. 4. Conclusions In this paper, we have considered the Markovian retrial queue with multiple types of outgoing calls and unreliable server. Using the asymptotic analysis method we have shown that the asymptotic behaviour of the number of customers in the orbit coincides with Gaussian distribution and in the limit condition → ∞, which means high rates of making outgoing calls, we have the mean 1 and variance 2.

About the authors

Anatoly A. Nazarov

National Research Tomsk State University

Author for correspondence.
Email: nazarov.tsu@gmail.com

Doctor of Technical Sciences, head of Department of Probability Theory and Mathematical Statistics

36, Lenina St., Tomsk, 634050, Russian Federation

Svetlana V. Paul

National Research Tomsk State University

Email: paulsv82@mail.ru

Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Probability Theory and Mathematical Statistics

36, Lenina St., Tomsk, 634050, Russian Federation

Olga D. Lizyura

National Research Tomsk State University

Email: oliztsu@mail.ru

Master’s Degree Student of Institute of Applied Mathematics and Computer Science

36, Lenina St., Tomsk, 634050, Russian Federation

References