Abstract
The Markov renewal process (MRP) of M/G/1 type has been used for modeling many complex queueing systems with correlated arrivals and the special types of transitions of the MRP process corresponds to the departures from the queueing system. It can be seen from the central limit theorem for regenerative process that the distribution of the number of transitions of MRP is asymptotically normal. Thus, the asymptotic mean and variance of the number of transitions of MRP can be used to estimate the number of departures in the queueing system modelled by MRP.
The aim of this paper is to present an algorithm for computing the asymptotic mean and variance for the number of leveldowntransitions in a Markov renewal process of M/G/1 type with finite level. The results are applied to the queueing system with finite buffer and correlated arrivals.
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References
Al Hanbali A, Mandjes A, Nazarathy Y, Whitt W (2011) The asymptotic variance of departures in critically loaded queues. Adv Appl Probab 43 (1):243–263
Angius A, Horváth A, Colledani M (2014) Moments of accumulated reward and completion time in Markovian models with application to unreliable manufacturing systems. Perform Eval 7576:69–88
Artalejo JR, GómezCorral A, He QM (2010) Markovian arrivals in stochastic modelling: a survey and some new results. Sort 34(2):101–144
Asmussen S, Bladt M (1994) Poisson’s equation for queues driven by a Markovian marked point process. Queueing Syst 17:235–274
Dallery Y, Gershwin B (1992) Manufacturing flow line systems: a review of models and analytical results. Queueing Syst 12:3–94
Ferng HW, Chang JF (2001) Departure processes of BMAP/G/1 queues. Queueing Syst 39:109–135
Gaver DJ (1962) A waiting line with interrupted service, including priorities. J R Stat Soc Ser B24:73–90
Heindl A, Telek M (2002) Output models of MAP/PH/1(/K) queues for an efficient network decomposition. Perform Eval 49:321–339
Horváth G, van Houdt B (2013) Departure process analysis of the multitype MMAP[K]/PH[K]/1 FCFS queue. Perform Eval 70:423–439
Lagershausena S, Tan B (2015) On the exact interdeparture and interstart time distribution of closed queueing networks subject to blocking. IIE Trans 47:673–692
Lee YH, Luh H (2006) Characterizing out processes of E_{m}/E_{k}/1 queues. Math Comput Model 44:771–789
Li J, Blumenfeld DE, Huang N, Alden JM (2009) Throughput analysis of production systems: recent advances and future topics. Int J Prod Res 47 (14):3823–3851
Liu Y, Wang P, Zhao YQ (2018) The variance constant for continuoustime level dependent quasibirthanddeath processes. Stoch Model 34(1):25–44
Lucantoni DM (1991) New results on the single server queue with a batch Markovian arrival process. Stoch Model 7:1–46
Lucantoni DM, MeierHellstern KS, Neuts MF (1990) A single server queue with server vacations and a class of nonrenewal arrival processes. Adv Appl Probab 22:676–705
Lucantoni DM, Neuts MF (1994) Some steadystate distributions for the MAP/SM/1 queue. Stoch Model 10(3):575–598
Narayana S, Neuts MF (1992) The first two moment matrices of the counts for the Markovian arrival process. Stoch Model 8(3):459–477
Nazarathy Y, Weiss G (2008) The asymptotic variance rate of the output process of finite capacity birthdeath queues. Queueing Syst 59(2):135–156
Neuts MF (1981) Matrixgeometric solutions in stochastic models, an algorithmic approach. Dover Publications, New York
Neuts MF (1989) Structured stochastic matrices of M/G/1 type and their applications. Marcel Dekker, New York
Shin YW (2019) Asymptotic mean and variance of departures in an M/G/1/K + 1 queue. Oper Res Lett 47:244–249
Tan B (2000) Asymptotic variance rate of the output in production lines with finite buffers. Ann Oper Res 93:385–403
Tan B (2013) Modeling and analysis of output variability in discrete material flow production systems. In: Tan B, Smith JM (eds) Handbook of Stochastic Models and Analysis of Manufacturing System Operations. Springer, New York, pp 287–311
Telek M, Pfening A (1996) Performance analysis of Markov reward models. Perform Eval 2728:1–18
Wolff RW (1989) Stochastic modeling and the theory of queues. Prentice Hall, Inc, Englewood Cliffs, NJ
Funding
This research was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Numbers NRF2018R1D1A1A09083352).
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Appendix: Uniformization Method for Computing A n, \(A^{\prime }_{n}\) and \(A^{\prime \prime }_{n}\)
Appendix: Uniformization Method for Computing A _{n}, \(A^{\prime }_{n}\) and \(A^{\prime \prime }_{n}\)
Let \(\theta =\max \limits _{1\le j\le m}\left \{ [D_{0}]_{jj}\right \}\) and
and for n = 0, 1, 2,⋯, k = 0, 1, 2,
By repeating the uniflormization arguments (Lucantoni 1991) shows that A_{n}, n = 0, 1, 2,⋯ are given by
where the matrices \(K^{(j)}_{n}\) are defined recursively as follows:

\(K^{(0)}_{0}=I\), \(K^{(0)}_{n}=0\), n ≥ 1,

For j = 1, 2,⋯,
$$ \begin{array}{@{}rcl@{}} K^{(j)}_{0}&=&K^{(j1)}_{0}P_{0},\\ K^{(j)}_{n}&=&\left\{\begin{array}{ll} K^{(j1)}_{n}P_{0}+K^{(j1)}_{n1}P_{1}, &1\le n\le j,\\ 0, & n\ge j+1. \end{array}\right. \end{array} $$
The matrices \(A^{\prime }_{n}\) and \(A^{\prime \prime }_{n}\) are obtained similarly by using γ_{n,1} and γ_{n,2} instead of γ_{n,0} in the formula A_{n}, respectively.
We present the formulae of γ_{n,k} for some distributions.

(i)
Deterministic distribution of service time with mean h_{1}:
$$ \gamma_{n,k}=e^{\theta h_{1}}\frac{\theta^{n}}{n!}h_{1}^{n+k}, n=0,1,2,\cdots, k=0,1,2. $$ 
(ii)
Gamma distribution Gam(a,b) of service time with mean h_{1} and SCV \({c_{s}^{2}}\): The parameters a and b are determined by
$$a=\frac{1}{{c_{s}^{2}}}, b={c_{s}^{2}} h_{1}$$and the LST of Gam(a,b) is H(s) = (1 + bs)^{−a}. Let \(\gamma _{k}(z)={\sum }_{n=0}^{\infty } \gamma _{n,k} z^{n}\), k = 0, 1, 2 and c(0) = 1, c(1) = a, c(2) = a(a + 1). Simple algebra yields
$$ \begin{array}{@{}rcl@{}} \gamma_{k}(z)&=&\frac{{\Gamma}(a+k)}{{\Gamma}(a)}\beta^{k}{\int}_{0}^{\infty} e^{\theta(1z)x}\frac{1}{{\Gamma}(a+k)}\frac{1}{b}e^{x/b}\left( \frac{x}{b}\right)^{a+k1} dx\\ &=&c(k)b^{k}\left( 1+b \theta(1z)\right)^{(a+k)}\\ &=&c(k)\frac{b^{k}}{(1+b\theta)^{a+k}} \left( 1\frac{b\theta}{1+b\theta}z\right)^{(a+k)}. \end{array} $$
Expanding the generating function γ_{k}(z), we have that
where
Thus for n = 0, 1, 2,⋯, k = 0, 1, 2,
As a special case, the exponential distribution with mean h_{1} is Gam(1,h_{1}) and hence the γ_{n,k} for exponential distribution of service time is
with c(0) = c(1) = 1 and c(2) = 2.
(iv) PH(β,S) distribution of service time : The LST of PH(β,S) is
where S^{0} = −Se. Note that
Thus for n = 0, 1, 2,⋯, k = 0, 1, 2,
(v) The γ_{n,k} for lognormal distribution LN(a,b) and Weibul distribution WB(a,b) of service time is obtained by numerical integration. The probability density function of LN(a,b) is
and the parameters a and b are determined by h_{1} and \({c_{s}^{2}}\) as follows
The probability density function of WB(a,b) is
and the parameters a and b are determined by h_{1} and \({c_{s}^{2}}\) as follows
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Shin, Y.W. An Algorithm for Asymptotic Mean and Variance for Markov Renewal Process of M/G/1 Type with Finite Level. Methodol Comput Appl Probab (2021). https://doi.org/10.1007/s1100902109846w
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Keywords
 First return time
 Fundamental period
 Asymptotic mean
 Asymptotic variance
 Markov renewal process of M/G/1 type
Mathematics Subject Classification (2010)
 60K20
 60K25