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.
Similar content being viewed by others
References
Aguir S, Karaesmen F, Aksin OZ, Chauvet F (2004) The impact of retrials on call center performance. OR Spectr 26:353–376
Aguir MS, Aksin OZ, Karaesmen F, Dallery Y (2008) On the interaction between retrials and sizing of call centers. Eur J Oper Res 191:398–408
Aksin Z, Armony M, Mehrotra V (2007) The modern call center: a multi-disciplinary perspective on operations management research. Prod Oper Manag 16:665–688
Apaolaza NM, Artalejo JR (2005) On the time to reach a certain orbit level in multi-server retrial queues. Appl Math Comput 168:686–703
Artalejo JR, Gomez-Corral A (2008) Retrial queueing systems: a computational approach. Springer, Berlin
Artalejo JR, Pla V (2009) On the impact of customer balking, impatience and retrials in telecommunication systems. Comput Math Appl 57:217–229
Artalejo JR (2010) Accessible bibliography on retrial queues: progress in 2000–2009. Math Comput Model 51:1071–1081
Baumann H, Sandmann W (2010) Numerical solution of level dependent quasi-birth-and-death processes. Procedia Comput Sci 1:1561–1569
Baumann H, Sandmann W (2012) Steady state analysis of level dependent quasi-birth-and-death processes with catastrophes. Comput Oper Res 39:413–423
Bright L, Taylor PG (1995) Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Stoch Model 11:497–525
Brown L, Gans N, Mandelbaum A, Sakov A, Shen H, Zeltyn S, Zhao L (2005) Statistical analysis of a telephone call center: a queueing-science perspective. J Am Stat Assoc 100:36–50
Chen Y, Chen P, Zhu Y (2016) Analysis of a call center with partial closing rules, feedback and impatient calls. Int J Appl Math 46:585–591
Chen P, Chen Y (2017) Analysis of a call center with impatient customers and repairable server. AMSE J-AMSE IIETA 54:127–135
Colladon AF, Naldi M, Schiraldi MM (2013) Quality management in the design of TLC call centres. Int J Eng Bus Manag 5:1–9
Diamond JE, Alfa AS (1998) The MAP/PH/1 retrial queue. Stoch Model 14:1151–1177
Essafi L, Bolch G (2005) Time dependent priorities in call centers. IJ Simul 6:32–38
Falin GI, Templeton JGC (1997) Retrial queues. Chapman and Hall, London
Gans N, Koole G, Mandelbaum A (2003) Telephone call centers: tutorial, review, and research prospects. Manuf Serv Oper Manag 5:79–141
Gomez-Corral A (2006) A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Ann Oper Res 141:163–191
Hashizume K, Phung-Duc T, Kasahara S, Takahasi Y (2012) Queueing analysis of internet - based call centers with intractive voice response and redial. In: Proceedings of 2012 IEEE 17th International Workshop on Computer Aided Modeling and Design of Communication Links and Networks (CAMAD) Barcelona: 373–377
Jouini O, Pot A, Koole G, Dallery Y (2010) Online scheduling policies for multiclass call centers with impatient customers. Eur J Oper Res 207:258–268
Jouini O, Koole G, Roubos A (2013) Performance indicators for call centers with impatient customers. IIE Trans 45:341–354
Khudyakov P, Feigin PD, Mandelbaum A (2010) Designing a call center with an IVR (Interactive Voice Response). Queueing Syst 66:215–237
Kim JW, Park SC (2010) Outsourcing strategy in two-stage call centers. Comput Oper Res 37:790–805
Kim C, Dudin A, Dudin S, Dudina O (2013) Tandem queueing system with impatient customers as a model of call center with interactive voice response. Perform Eval 70:440–453
Kim C, Klimenok VI, Dudin AN (2016) Priority tandem queueing system with retrials and reservation of channels as a model of call center. Comput Ind Eng 96:61–71
Klimenok V, Savko R (2013) A retrial tandem queue with two types of customers and reservation of channels. In: Proceedings of modern probabilistic methods for analysis of telecommunication networks, (BWWQT 2013), CCIS, vol 356, pp 105–114
Koole G, Mandelbaum A (2002) Queueing models of call centers: an introduction. Ann Oper Res 113:41–59
Lebedev EA (2002) On the first passage time of removing level for retrial queues. Rep Nat Acad Sci Ukraine 3:47–50
Li N a, Yu X, Matta A (2017) Modelling and workload reallocation of call centres with multi-type customers. Int J Prod Res 55:5664–5680
Nazarov A, Phung-Duc T, Paul S (2018) Unreliable single-server queue with two-way communication and retrials of blocked and interrupted calls for cognitive radio networks. In: Proceedings of distributed computer and communication networks (DCCN 2018), CCIS, vol 919, pp 276–287
Neuts MF (1981) Matrix-geometric solutions in stochastic models – an algorithmic approach. The Johns Hopkins University Press, Baltimore
Senthil Kumar M, Dadlani A, Kim K (2018) Performance analysis of an unreliable M/G/1 retrial queue with two-way communication. Oper Res Int J, 1–14
Srinivasan R, Talim J, Wang J (2004) Performance analysis of a call center with interactive voice response units. Sociedad de Estadıstica e Investigacion Operativa Top 12:91–110
Tweedie RL (1975) Sufficient conditions for regularity, recurrence and ergodicity of Markov processes. Math Proc Cambridge Philos Soc 78:125–136
Wang J, Srinivasan R (2008) Staffing a call center with interactive voice response units and impatient calls. In: Proceedings of international conference on service operations and logistics and informatics. IEEE, pp 514–519
Zhang H (2010) Performance analysis in call centers with IVR and impatient customers. Journal of East China Normal University (Natural Sc 2010), 69–78
Acknowledgments
The authors are grateful to the anonymous referees and associate editor for their critical reading of the manuscript, fruitful suggestions and constructive comments that have improved the presentation and quality of this manuscript. This research work is supported in part by Department of Science and Technology, Government of India under grant INT/RUS/RFBR/377.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Proof of Theorem 1
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
of order (M + 1)(J1 + 1) for upper diagonal, diagonal and lower diagonal matrices, respectively, where ΔM = −diag(AM,1) is the diagonal matrix of AM,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
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
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
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
whence
where \(\mathbf {A}_{M}(z)=\mathbf {A}_{M,0}+z\mathbf {A}_{M,1}+z^{2}\mathbf {A}_{M,2}\) and 0T 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
where we take a ∈ (0, 1) and the column sub-vector wm(z) consisting of the first (m + 1)(J1 + 1) elements of \(\mathbf {V}_M^T(z)\) is given by
Based on which, we now construct the column vector, \(\boldsymbol {\phi }_n^T\), of entries \(\boldsymbol {\phi }_{m,n}^T\), for n ≥ 0, as
and the non-negative vector-valued test (or Lyapunov) function
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
holds for all but a finite number n ≥ 0 and for some 𝜖 > 0.
Now, we have for n = 1, 2, 3,⋯ ,
whence, for n ≥ 1, after some algebraic calculation,
where
m = 1, 2,⋯ ,M − 1,
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,
and
Thus there exists an integer n0 and some 𝜖 > 0 such that
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). □
Rights and permissions
About this article
Cite this article
Kumar, B.K., Sankar, R., Krishnan, R.N. et al. Performance Analysis of Multi-processor Two-Stage Tandem Call Center Retrial Queues with Non-Reliable Processors. Methodol Comput Appl Probab 24, 95–142 (2022). https://doi.org/10.1007/s11009-020-09842-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-020-09842-6