The paper explores parameter estimation in a Markovian queuing model with Poisson arrival rates and state-dependent service, where the server’s efficiency varies based on queue length. The focus is on estimating traffic intensity () using a Bayesian approach with various asymmetric loss functions. This estimation process involves the exploration of different prior distributions, including beta, truncated gamma, and uniform, to assess traffic intensity. The study utilizes Markov Chain Monte Carlo (MCMC) simulations for robust traffic intensity estimation. Credible intervals are introduced alongside point estimates to account for uncertainties, providing a more comprehensive understanding. The analysis extends to incorporate posterior risk, considering both estimates and associated uncertainties in the assessment process. In the end, the analysis includes an evaluation of real-life data.
AlmeidaM. A.CruzF. R. (2018). A note on Bayesian estimation of traffic intensity in single-server markovian queues. Communications in Statistics-Simulation and Computation, 47(9), 2577–2586.
2.
ArmeroC.BayarriM. (1994a). Prior assessments for prediction in queues. Journal of the Royal Statistical Society Series D: The Statistician, 43(1), 139–153.
3.
ArmeroC.BayarriM. J. (1994b). Bayesian prediction in M/M/1 queues. Queueing Systems, 15, 401–417.
4.
BasawaI.PrabhuN. (1981). Estimation in single server queues. Naval Research Logistics Quarterly, 28(3), 475–487.
5.
BenešV. E. (1957). A sufficient set of statistics for a simple telephone exchange model. Bell System Technical Journal, 36(4), 939–964.
6.
BuraG. S.SharmaH. (2024). Maximum likelihood and bayesian estimation on m/m/1 queueing model with balking. Communications in Statistics-Theory and Methods, 53, 5117–5145.
7.
BuraG. S.SharmaH. (2025). Bayesian analysis of single server markovian queueing model with balking under asymmetric loss functions. Communications in Statistics-Simulation and Computation, 54, 144–159.
8.
CalabriaR.PulciniG. (1994). An engineering approach to bayes estimation for the weibull distribution. Microelectronics Reliability, 34(5), 789–802.
9.
ChapmanD. G. (1956). Estimating the parameters of a truncated gamma distribution. The Annals of Mathematical Statistics, 27(2), 498–506.
10.
ChoudhuryA.BasakA. (2018). Statistical inference on traffic intensity in an M/M/1 queueing system. International Journal of Management Science and Engineering Management, 13(4), 274–279.
11.
ChoudhuryA.BorthakurA. C. (2007). Statistical inference in M/M/1 queues: A Bayesian approach. American Journal of Mathematical and Management Sciences, 27(1-2), 25–41.
12.
ChoudhuryA.BorthakurA. C. (2008). Bayesian inference and prediction in the single server markovian queue. Metrika, 67, 371–383.
13.
ClarkeA. B., et al. (1957). Maximum likelihood estimates in a simple queue. The Annals of Mathematical Statistics, 28(4), 1036–1040.
14.
CruzF. R.AlmeidaM. A.D’AngeloM. F.van WoenselT. (2018). Traffic intensity estimation in finite markovian queueing systems. Mathematical Problems in Engineering, 2018(2), 1–15.
15.
CruzF. R.QuininoR.HoL. L. (2017). Bayesian estimation of traffic intensity based on queue length in a multi-server M/M/S queue. Communications in Statistics-Simulation and Computation, 46(9), 7319–7331.
16.
DaveU.ShahY. (1980). Maximum likelihood estimates in a M/M/2 queue with heterogeneous servers. Journal of the Operational Research Society, 31(5), 423–426.
17.
DeepthiV.JoseJ. K. (2020). Bayesian estimation of an M/M/R queue with heterogeneous servers using markov chain monte carlo method. Stochastics and Quality Control, 35(2), 57–66.
18.
DeGrootM. H. (2005). Optimal Statistical Decisions. John Wiley & Sons.
19.
GoyalT. L.HarrisC. M. (1972). Maximum-likelihood estimates for queues with state-dependent service. Sankhyā: The Indian Journal of Statistics, Series A, 34(1), 65–80.
20.
JabaraliA.KannanK. S. (2016). Single server queueing model with gumbel distribution using Bayesian approach. Hacettepe Journal of Mathematics and Statistics, 45(4), 1275–1296.
21.
LillieforsH. W. (1966). Letter to the editor-some confidence intervals for queues. Operations Research, 14(4), 723–727.
22.
MuddapurM. (1972). Bayesian estimates of parameters in some queueing models. Annals of the Institute of Statistical Mathematics, 24(1), 327–331.
23.
MukherjeeS.ChowdhuryS. (2010). Bayes estimation of measures of effectiveness in an M/M/1 queueing model. Calcutta Statistical Association Bulletin, 62(1-2), 97–108.
24.
NorstromJ. G. (1996). The use of precautionary loss functions in risk analysis. IEEE Transactions on Reliability, 45(3), 400–403.
25.
RubinD. B. (1988). Using the sir algorithm to simulate posterior distributions. In Bayesian statistics 3. Proceedings of the third Valencia international meeting, 1-5 June 1987 (pp. 395–402). Clarendon Press.
26.
ShortleJ. F.ThompsonJ. M.GrossD.HarrisC. M. (2018). Fundamentals of Queueing Theory. 5th ed, Wiley.
27.
SmithA. F.GelfandA. E. (1992). Bayesian statistics without tears: A sampling–resampling perspective. The American Statistician, 46(2), 84–88.
28.
ThiruvaiyaruD.BasawaI. V. (1992). Empirical bayes estimation for queueing systems and networks. Queueing Systems, 11(3), 179–202.
29.
WangK.-H.ChenS.-C.KeJ.-C. (2004). Maximum likelihood estimates and confidence intervals of an M/M/R/N queue with balking and heterogeneous servers. RAIRO-Operations Research, 38(3), 227–241.
30.
WangT.-Y.KeJ.-C.WangK.-H.HoS.-C. (2006). Maximum likelihood estimates and confidence intervals of an M/M/R queue with heterogeneous servers. Mathematical Methods of Operations Research, 63(2), 371–384.
31.
ZellnerA. (1986). Bayesian estimation and prediction using asymmetric loss functions. Journal of the American Statistical Association, 81(394), 446–451.
