Multi-server queueing system with a generalized phase-type service time distribution as a model of call center with a call-back option

Abstract

A multi-server queueing system with a Markovian arrival process and finite and infinite buffers to model a call center with a call-back option is investigated. If all servers are busy during the customer arrival epoch, the customer may leave the system forever or move to the buffer (such a customer is referred to as a real customer), or, alternatively, request for call-back (such a customer is referred to as a virtual customer). During a waiting period, a real customer can be impatient and may leave the system without service or request for call-back (becomes a virtual customer). The service time of a customer and the dial time to a virtual customer for a server have a phase-type distribution. To simplify the investigation of the system we introduce the notion of a generalized phase-type service time distribution. We determine the stationary distribution of the system states and derive the Laplace–Stieltjes transforms of the sojourn and waiting time distributions for real and virtual customers. Some key performance measures are calculated and numerical results are presented.

Appendices

Appendix 1

Let us present some examples of using the generalized phase-type distribution in an investigation of queues with heterogeneous customers and different service time distributions for each type of customers.

• (i) If we consider the system with $K$ types of customers and the service time of type $k$ customer,$\;\; k=\overline{1,K},$ has an exponential distribution with the parameter $\mu _k,$ it is reasonable to consider a generalized $\textit{PH}$ distribution with an irreducible representation $({\varvec{\beta }^{(1)}},\ldots ,{\varvec{\beta }^{(K)}}, S)$ where $\varvec{\beta }^{(k)},\,k=\overline{1,K},$ is the vector of size $K$ with all zero entries except the $k$th entry, which is equal to 1, and the matrix $S$ of the form $S=-\mathrm{diag}\{\mu _1,\ldots ,\mu _K\}$.

• (ii) If we consider the system with $K$ types of customers and the service time of type $k$ customer,$\;\; k=\overline{1,K},$ has a phase-type distribution with an irreducible representation $( \varvec{\hat{\beta }}^{(k)}, S^{(k)})$ and the finite state space $\{1,\ldots , M_k, M_k+1\},$ it is reasonable to consider a generalized $\textit{PH}$ distribution with an irreducible representation $({\varvec{\beta }^{(1)}},\ldots ,{\varvec{\beta }^{(K)}}, S)$ where $\varvec{\beta }^{(k)}=(\mathbf{0}_{\sum \nolimits _{l=1}^{k-1} M_l}, \varvec{\hat{\beta }}^{(k)},\mathbf{0}_{\sum \nolimits _{l=k+1}^{K} M_l}),\,k=\overline{1,K},$ and the matrix $S$ of the form $S=\mathrm{diag}\{S_1,\ldots ,S_K\}$.

Appendix 2

In Appendix 2, we prove the assertion that the use of the generalized phase-type with the state space $({1,\ldots ,M_1+\cdots +M_l+1})$ instead of the consideration of $l$ phase type service processes with dimensions $M_n+1, \,n=\overline{1,l}$ of the state space does not change the dimension of the stochastic process that describes the behavior of the system under consideration.

Let us assume that $l$ different types of customers are serviced in any of $N$ servers. The service time of type $n$ customer has a phase-type distribution with the dimension of the space of the non-absorbing states $M_n,\;n=\overline{1,l}.$ We use approach by V. Ramaswami and D.M. Lucantoni, so, the dimension of components of the Markov chain that describes the service process of all types of customers $K_1$ is equal to

$$\begin{aligned} K_1&= \sum \limits _{k_l=0}^N\sum \limits _{k_{l-1}=0}^{N-k_l} \dots \sum \limits _{k_2=0}^{N-({k_l+k_{l-1}+\cdots +k_3})} \left( \begin{array}{c} {k_l+M_l-1}\\ {M_l-1} \end{array}\right) \times \cdots \\&\times \left( \begin{array}{c} {k_2+M_2-1}\\ {M_2-1}\end{array}\right) \left( \begin{array}{c} {N-(k_l+k_{l-1}+\cdots +k_2)+M_1-1}\\ {M_1-1} \end{array}\right) , \end{aligned}$$

where the summation index $k_n$ has meaning of the number of type $n$ customers presenting in the system.

If we describe processing of customers by a generalized phase-type service process, the dimension of its state space is $K_2=\left( \begin{array}{c}{N+M_1+\cdots +M_l-1}\\ {M_1+\cdots +M_l-1}\end{array}\right)$.

So, we assert that $K_1=K_2$ for any $N$ and $l.$

Let us first consider the case $l=2$, i.e., there are two types of customers and the service time of type $n$ customer has a phase-type distribution with dimension of the space of the non-absorbing states $M_n, n=1,2.$ In this case the dimension $K_1$ is given by formulae $K_1=\sum \nolimits _{k_2=0}^N\left( \begin{array}{c}{k_2+M_2-1}\\ {M_2-1}\end{array}\right) \left( \begin{array}{c}{N-k_2+M_1-1}\\ {M_1-1}\end{array}\right)$. Using the following formulae

$$\begin{aligned} \sum \limits _{k=0}^N\left( \begin{array}{c}{k+a}\\ {a}\end{array}\right) \left( \begin{array}{c}{b-k}\\ {b-n}\end{array}\right) =\left( \begin{array}{c}{a+b+1}\\ {n}\end{array}\right) \end{aligned}$$

(11)

and setting $a=M_2-1,\,b=N+M_1-1$ it is easy to verify that $K_1=\left( \begin{array}{c}{N+M_1+M_2-1}\\ {N}\end{array}\right) =\left( \begin{array}{c}{N+M_1+M_2-1}\\ {M_1+M_2-1}\end{array}\right) =K_2$. So, for $l=2$ the dimensions $K_1$ and $K_2$ are equal.

Let us assume that $K_1=K_2$ for the case $l,\;l \ge 2,$ types of customers, i.e.,

$$\begin{aligned}&\sum \limits _{k_l=0}^N\sum \limits _{k_{l-1}=0}^{N-k_l}\cdots \sum \limits _{k_2=0}^{N-({k_l+k_{l-1}+\cdots +k_3})} \left( \begin{array}{c}{k_l+M_l-1}\\ {M_l-1}\end{array}\right) \times \cdots \times \nonumber \\&\quad \times \left( \begin{array}{c}{k_2+M_2-1}\\ {M_2-1}\end{array}\right) \left( \begin{array}{c}{N-(k_l+k_{l-1}+\cdots +k_2)+M_1-1}\\ {M_1-1}\end{array}\right) \nonumber \\&=\left( \begin{array}{c}{N+M_1+\cdots +M_l-1}\\ {M_1+\cdots +M_l-1}\end{array}\right) , \end{aligned}$$

(12)

and prove that $K_1=K_2$ for the case $l+1$ types of customers as well.

$$\begin{aligned} K_1&= \sum \limits _{k_{l+1}=0}^N\sum \limits _{k_{l}=0}^{N-k_{l+1}}\dots \sum \limits _{k_2=0}^{N-({k_{l+1}+k_{l}+\cdots +k_3})} \left( \begin{array}{c}{k_{l+1}+M_{l+1}-1}\\ {M_{l+1}-1}\end{array}\right) \times \cdots \times \\&\times \left( \begin{array}{c}{k_2+M_2-1}\\ {M_2-1}\end{array}\right) \left( \begin{array}{c}{N-(k_{l+1}+k_{l}+\cdots +k_2)+M_1-1}\\ {M_1-1}\end{array}\right) =[(A2)]\\&= \sum \limits _{k_{l+1}=0}^N\left( \begin{array}{c}{k_{l+1}+M_{l+1}-1}\\ {M_{l+1}-1}\end{array}\right) \left( \begin{array}{c}{N-k_{l+1}+M_{1}+\cdots +M_l-1}\\ {M_{1}+\cdots +M_l-1}\end{array}\right) =[(A1)]\\&= \left( \begin{array}{c}{N+M_{1}+\cdots +M_{l+1}-1}\\ {M_{1}+\cdots +M_{l+1}-1}\end{array}\right) =K_2. \end{aligned}$$

So, by induction the assertion is proved.