Performance Analysis of Multi-processor Two-Stage Tandem Call Center Retrial Queues with Non-Reliable Processors

Abstract

We analyze a multi-processor two-stage tandem call center retrial queueing network in which the processors are subject to active breakdowns and repairs at stage-I. A level-dependent quasi-birth-and-death (LDQBD) process is formulated and a sufficient condition for ergodicity of the system is discussed. Under the stability condition, the stationary distribution of the number of calls in the system, the mean number of calls in the orbit, the mean waiting time of calls in the orbit and the mean busy period of the system along with other descriptors of the system are determined by using the matrix-analytic techniques. Besides, the availability analysis of the processors is also studied. Further, we have also discussed the characteristics of the first-passage time to reach the orbit critical level, the number of calls served in the call center during this period and their corresponding moments. Finally, extensive numerical results are presented to highlight the impact of the system parameters on the performance measures of the two-stage tandem call center retrial queueing system under investigation.

Appendix: Proof of Theorem 1

Proof

To prove the statement of the theorem, we mostly make use of the methodology of Diamond and Alfa (1998) and approach by Tweedie (1975). To this end, we now construct a discrete-time homogeneous LIQBD process with block square matrices

$$\overline{\mathbf{A}}_{M,0} = \mathbf{\Delta}_{M}^{-1}\mathbf{A}_{M,0}, \overline{\mathbf{A}}_{M,1} = \mathbf{\Delta}_{M}^{-1}\mathbf{A}_{M,1}+\mathbf{I}_{(M+1)(J_{1}+1)} \text{and} \overline{\mathbf{A}}_{M,2} = \mathbf{\Delta}_{M}^{-1}\mathbf{A}_{M,2}, $$

of order (M + 1)(J₁ + 1) for upper diagonal, diagonal and lower diagonal matrices, respectively, where Δ_M = −diag(A_M,1) is the diagonal matrix of A_M,1. Thus the transition probability matrix $\overline {\mathbf {A}}_{M}=\overline {\mathbf {A}}_{M,0}+\overline {\mathbf {A}}_{M,1}+\overline {\mathbf {A}}_{M,2}$ of order (M + 1)(J₁ + 1) is stochastic and irreducible with state space $\boldsymbol {\overline {\chi }}$. The corresponding stationary distribution $\mathbf {\overline {Y}}_{M}$ of $\mathbf {\overline {A}}_{M}$ is obtained as

$$\overline{\mathbf{Y}}_{M}=\left[\left( \mathbf{Y}_{M} A_{M} \mathbf{e}_{(M+1)(J_{1}+1)\times 1}^{T}\right)^{-1}\mathbf{Y}_{M}{\Delta}_{M}\right]_{1\times(M+1)(J_{1}+1)}.$$

By using ρ < 1, i.e., $\mathbf {Y}_{M}\mathbf {A}_{M,0} \mathbf {e}_{(M+1)(J_{1}+1)\times 1}^{T}<\mathbf {Y}_{M}\mathbf {A}_{M,2} \mathbf {e}_{(M+1)(J_{1}+1)\times 1}^{T}$, we have

$$\overline{\mathbf{Y}}_{M}\overline{\mathbf{A}}_{M,0} \mathbf{e}_{(M+1)(J_{1}+1)\times1}^{T}<\overline{\mathbf{Y}}_{M}\overline{\mathbf{A}}_{M,2} \mathbf{e}_{(M+1)(J_{1}+1)\times1}^{T}$$

which guarantees the ergodicity of the discrete-time LIQBD process.

We now define an irreducible and non-negative matrix $\overline {\mathbf {A}}_{M}(z)=\overline {\mathbf {A}}_{M,0}+z\overline {\mathbf {A}}_{M,1}+z^{2}\overline {\mathbf {A}}_{M,2}$ for 0 ≤ z ≤ 1 and denote the spectral radius of $\overline {\mathbf {A}}_{M}(z)$ as $\chi (z)=sp(\mathbf {\overline {A}}_{M}(z))$. The rate matrix of the discrete time LIQBD process is denoted by R which is the minimal non-negative solution of the matrix quadratic equation

$$\overline{\mathbf{A}}_{M,0}+\mathbf{R} \overline{\mathbf{A}}_{M,1}+\mathbf{R^{2}} \overline{\mathbf{A}}_{M,2}=\mathbf{R}.$$

If $\overline {\eta }=sp(\mathbf {R})$ is the spectral radius of R, then $\overline {\eta }<1$ as ρ < 1. From the discussion in the proof of Lemma 1.3.4 in Neuts (1981), we have χ(z) < z for $\overline {\eta }<z<1$. Let $\mathbf {V}_{M}^{T}(z)$ be the right eigenvector of dimension (M + 1)(J₁ + 1) × 1 of $\overline {\mathbf {A}}_{M}(z)$ with strictly positive entries associated with χ(z), so that

$$ \overline{\mathbf{A}}_{M}(z) \mathbf{V}_{M}^{T}(z)=\chi(z) \mathbf{V}_{M}^{T}(z) < z \mathbf{V}_{M}^{T}(z), $$

whence

$$ \mathbf{A}_{M}(z) \mathbf{V}_{M}^{T}(z) < \textbf{0}^{T} \quad \text{for}\quad \overline{\eta} < z < 1, $$

where $\mathbf {A}_{M}(z)=\mathbf {A}_{M,0}+z\mathbf {A}_{M,1}+z^{2}\mathbf {A}_{M,2}$ and 0^T is zero (M + 1)(J₁ + 1) × 1 - dimensional column vector.

Besides, for $z\in (\overline {\eta },1),$ let us define a non-negative column vector, $\boldsymbol {\phi }_{m,n}^{T}$, of dimension (m + 1)(J₁ + 1) × 1 as

$$\boldsymbol{\phi}_{m,n}^{T}=z^{-n}\left[z^{M-m} \mathbf{w}_{m}(z) + a \mathbf{e}_{(m+1)(J_{1}+1)\times 1}^{T} \right]\quad \text{for} 0 \leq m \leq M \text{and} n \geq 1,$$

where we take a ∈ (0, 1) and the column sub-vector w_m(z) consisting of the first (m + 1)(J₁ + 1) elements of $\mathbf {V}_M^T(z)$ is given by

$$\mathbf{w}_{m}(z)=\mathbf{E}_{m,m+1} \mathbf{E}_{m+1,m+2} {\cdots} \mathbf{E}_{M-1,M}\mathbf{V}_{M}^{T}(z)\quad \text{for} m=0,1,2,\cdots,M-1, $$

$$\mathbf{w}_{M}(z)=\mathbf{V}_{M}^{T}(z).$$

Based on which, we now construct the column vector, $\boldsymbol {\phi }_n^T$, of entries $\boldsymbol {\phi }_{m,n}^T$, for n ≥ 0, as

$$\boldsymbol{\phi}_{n}^{T} = z^{-n} \left[ \left( \begin{array}{c} z^{M} \mathbf{w}_{0}(z) \\ z^{M-1}\mathbf{w}_{1}(z) \\ z^{M-2}\mathbf{w}_{2}(z) \\ {\vdots} \\ z^{M-m}\mathbf{w}_{m}(z) \\ \vdots\\ z \mathbf{w}_{M-1}(z)\\ \mathbf{w}_{M}(z) \end{array} \right)_{{\Gamma}_{M}^{(n)}\times1}+a \mathbf{e}_{{\Gamma}_{M}^{(n)}\times1}^{T} \right] = \left[ \begin{array}{c} \boldsymbol{\phi}_{0,n}^{T} \\ \boldsymbol{\phi}_{1,n}^{T} \\ \boldsymbol{\phi}_{2,n}^{T}\\ \vdots\\ \boldsymbol{\phi}_{m,n}^{T}\\ \vdots\\ \boldsymbol{\phi}_{M-1,n}^{T} \\ \boldsymbol{\phi}_{M,n}^{T} \end{array} \right]_{{\Gamma}_{M}^{(n)}\times1,}$$

and the non-negative vector-valued test (or Lyapunov) function

$$\boldsymbol{\phi}^{T}=\left[\boldsymbol{\phi}_{0}^{T}, \boldsymbol{\phi}_{1}^{T}, \boldsymbol{\phi}_{2}^{T},\cdots,\boldsymbol{\phi}_{n}^{T},{\cdots} \right]_{.}^{T}$$

It is observed that each element of $\boldsymbol {\phi }_{n}^{T}$ is bounded below for all n ≥ 0.

In order to prove the ergodicity, it is enough to prove that

$$\mathbf{Q}\boldsymbol{\phi}^{T} \leq -\epsilon \textbf{e}^{T}$$

holds for all but a finite number n ≥ 0 and for some 𝜖 > 0.

Now, we have for n = 1, 2, 3,⋯ ,

$$(\mathbf{Q}\boldsymbol{\phi}^{T})_{n}= \mathbf{A}_{n,2}\boldsymbol{\phi}_{n-1}^{T}+\mathbf{A}_{n,1}\boldsymbol{\phi}_{n}^{T}+\mathbf{A}_{n,0}\boldsymbol{\phi}_{n+1}^{T},$$

whence, for n ≥ 1, after some algebraic calculation,

$$ {(\mathbf{Q}\boldsymbol{\phi}^{T})_{n}} = z^{-(n+1)}\left[ \left( \begin{array}{c} z^{M} \mathbf{g}_{0}^{(n)}(z) \\ z^{M-1}\mathbf{g}_{1}^{(n)}(z) \\ z^{M-2}\mathbf{g}_{2}^{(n)}(z)\\ {\vdots} \\ z^{M-m}\mathbf{g}_{m}^{(n)}(z)\\ \vdots\\ z^{2}\mathbf{g}_{M-2}^{(n)}(z) \\ z \mathbf{g}_{M-1}^{(n)}(z)\\ \mathbf{g}_{M}^{(n)}(z) \end{array} \right)+a(1-z) \left( \begin{array}{c} \mathbf{0}_{{\gamma_{0}^{(n)}}}^{T}\mathbf{e}_{{\gamma_{0}^{(n)}}}^{T}-n\nu z \mathbf{e}_{{\gamma_{0}^{(n)}}}^{T} \\ \mathbf{A}_{n,0}^{1,2}\mathbf{e}_{{\gamma_{0}^{(n)}}}^{T}-n\nu z \mathbf{e}_{{\gamma_{1}^{(n)}}}^{T}\\ \mathbf{A}_{n,0}^{2,2}\mathbf{e}_{{\gamma_{1}^{(n)}}}^{T}-n\nu z \mathbf{e}_{{\gamma_{2}^{(n)}}}^{T}\\ \vdots\\ \mathbf{A}_{n,0}^{m,2}\mathbf{e}_{{\gamma_{(m-1)}^{(n)}}}^{T}-n\nu z \mathbf{e}_{{\gamma_{m}^{(n)}}}^{T}\\ \vdots\\ \mathbf{A}_{n,0}^{M-2,2}\mathbf{e}_{{\gamma_{M-3}^{(n)}}}^{T}-n\nu z \mathbf{e}_{{\gamma_{M-2}^{(n)}}}^{T}\\ \mathbf{A}_{n,0}^{M-1,2}\mathbf{e}_{{\gamma_{M-2}^{(n)}}}^{T}-n\nu z \mathbf{e}_{{\gamma_{M-1}^{(n)}}}^{T}\\ \mathbf{A}_{n,0}^{M,2}\mathbf{e}_{{\gamma_{M-1}^{(n)}}}^{T}+\mathbf{A}_{n,0}^{M,1} \mathbf{e}_{{\gamma_{M}^{(n)}}}^{T} \end{array} \right) \right],$$

where

$$\mathbf{g}_{0}^{(n)}(z)= \mathbf{A}_{n,1}^{0,0} \mathbf{w}_{1}(z) + z (\mathbf{A}_{n,1}^{0,1}+n \nu \mathbf{I}) \mathbf{w}_{0}(z),$$

$$\mathbf{g}_{m}^{(n)}(z)= \mathbf{A}_{n,1}^{m,0} \mathbf{w}_{m+1}(z) + z [(\mathbf{A}_{n,1}^{m,1}+n \nu \mathbf{I}) \mathbf{w}_{m}(z)+\mathbf{A}_{n,0}^{m,2}\mathbf{w}_{m-1}(z)]+z^{2} \mathbf{A}_{n,1}^{m,2}\mathbf{w}_{m-1}(z),$$

m = 1, 2,⋯ ,M − 1,

$$\mathbf{g}_{M}^{(n)}(z)=[\mathbf{A}_{M,0}+z\mathbf{A}_{M,1}+z^{2}\mathbf{A}_{M,2}]\mathbf{V}_{M}^{T}(z)$$

$$= \mathbf{A}_{M}(z) \mathbf{V}_{M}^{T}(z)$$

with $\mathbf {e}_{\gamma _m^{(n)}}^T$ is a column vector of ones of dimension $\gamma _m^{(n)}\times 1$ and I is the identity matrix of appropriate order.

Since $\mathbf {A}_{M}(z) \mathbf {V}_{M}^{T}(z) < \mathbf {0}_{\gamma _{M}^{(n)}\times 1}^{T}$ and $(1-z)(\mathbf {A}_{n,0}^{M,2} \mathbf {e}_{\gamma _{M-1}^{(n)}}^{T}+\mathbf {A}_{n,0}^{M,1}\mathbf {e}_{\gamma _{M}^{(n)}}^{T})>\mathbf {0}_{\gamma _{M}^{(n)}\times 1}^{T}$ for all $z\in (\overline {\eta },1),$ we can select a ∈ (0, 1) so that for n ≥ 0,

$$(\mathbf{Q} \boldsymbol{\phi}^{T})_{m,n} \leq -\epsilon \mathbf{e}^{T} \text{for} m=M$$

and

$$(\mathbf{Q} \boldsymbol{\phi}^{T})_{m,n} \rightarrow -\infty \quad as n \rightarrow \infty \text{for} m=0,1,2, \cdots,M-1.$$

Thus there exists an integer n₀ and some 𝜖 > 0 such that

$$(\mathbf{Q} \boldsymbol{\phi}^{T})_{m,n} \leq -\epsilon \mathbf{e}^{T} \text{for} n\geq n_{0}.$$

Hence, we conclude that the process {X(t); t ≥ 0} is regular and ergodic according to Tweedie (1975) or statement 8, pp. 97 in Falin and Templeton (1997). □