In this note, we consider discrete-time finite Markov Chains and assume that they are only partly observed. We obtain finite-dimensional normalized filters for basic statistics associated with such **processes**. Recursive equations for these filters are de- rived by means of simple computations involving conditional expectations. An appli- cation to the estimation of parameters for the so-called discrete-time **Batch** **Markovian** **Arrival** **Processes** is outlined.

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state 0, which means that the **arrival** distribution is arbitrary when nobody is in the queue. Then A. Y. Chen and E. Renshaw (1997) (2004) established the possibility to let the queue system idle(i.e. nobody is in the queue) at any time. However, all the models discussed before are only the simple queue (i.e. only one person arrives or leaves at the same moment), which eliminates the cases such practical cases as the waiting for the lift and the **arrival** of the passengers in the aircraft. Since M. F. Neuts (1979) introduced versatile **Markovian** **arrival** **processes** by using several kinds of **batch**-**arrival** process and M. L. Chaudhry and J. G. C. Templeton (1983) discussed the first course of bulk queues, the theory of **batch** **arrival** and bulk service have been well developed until now. For example, we can see the most recent result from C. Armero and D. Conesa (2000), R. Arumuganathan and K. S. Ramaswami (2005), S. H. Chang, D. W. Choi and T. S. Kim (2004), D. Fakinos (1991), L. Srinivasan, N. Renganathan and R. Kalyanaraman (2002), U. Sumita and Y. Masuda (1997) and P. V. Ushakumari and A. Krishnamoorthy (1998).

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The study of the of the time-dependent queue length distribution of M/G/1 and GI/M/1 type of queues via Markov regenerative **processes** is not new. It is known (see for example Cinlar 8] and Kulkarni 10]) that the queue length distribution at an arbitrary time instant is a Markov regenerative process (or semi-regenerative according to Cinlar's notation). How- ever, the numerical computation of the queue length distribution gives rise to a system of integral equations that are in general hard to solve. The focus of this chapter is therefore to develop numerical algorithms, especially when the non-**Markovian** parameters are deter- ministic, to numerically compute the queue length distribution of M/G/1 and GI/M/1 type of queues. As a further extension, we allow more general **arrival** or service time **processes** such as the **Batch** **Markovian** **arrival** process (BMAP). As mentioned earlier, we also consider multi-class queueing systems.

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In an attempt to examine the effect of dependencies in the **arrival** process on the steady state queue length process in single server queueing models with exponential service time distribution, four different models for the **arrival** process, each with marginally distributed exponential inter- arrivals to the queueing system, are considered. Two of these models are based upon the upper and lower bounding joint distribution functions given by the Fréchet bounds for bivariate distributions with specified marginals, the third is based on Downton’s bivariate exponential distribution and fourthly the usual M/M/1 model. The aim of the paper is to compare conditions for stability and explore the queueing behaviour of the different models.

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W e consider a multi-server infinite capacity **Markovian** feedback queue with reneging, balking and retention of reneged customers in which the inter-**arrival** and service times follow exponential distribution. The reneging times are assumed to be exponentially distributed. After the completion of service, each customer may rejoin the system as a feedback customer for receiving another regular service with a certain probability. A reneged customer can be retained in many cases by employing certain convincing mechanisms to stay in queue for completion of service. Numerical analysis, cost-profit analysis and optimization of the cost function using simulated annealing method are carried out. The steady-state solution of the model is obtained iteratively. Some performance measures are also derived.

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Composition is not shown in this diagram, but could be easily calculated based on the mass balance in each time inter.val Both the TGD and OTGD will be used to design the batch M[r]

Chapter 3 fleshes out the theory behind the intermediary **processes** introduced in Chapter 2. Section 3.2 contains a proof that the number of iterations before convergence for the averaged range process defined in Chapter 2 has the same distribution as the number of iterations before convergence for the original algorithm. This stage of the approximation framework can thus be reached with no loss of information concerning convergence rates. Section 3.3 demonstrates the existence of the final intermediary process defined in Chapter 2, the asymptotic averaged range process. A subsection developing preliminary Markov chain theory precedes the existence proof, which is presented initially in the case where the domain transition matrix on suboptimal states is both irreducible and acyclic and then in the more general case where only irreducibility is assumed. The convergence behaviour of the asymptotic averaged range process is then examined in Section 3 .4. The expected convergence time is different to the true value for this approximation; however, in many cases the error is very small.

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In this paper, a multi-server, finite Capacity, interdependent queueing model, controllable **arrival** rates with discouraged arrivals and retention of reneged customers is considered. The steady state probabilities of system size are derived explicitly. The effect of the probability of customer’s retention on the expected system size has been studied. The analytical results are numerically illustrated and relevant conclusions are presented.

of early works on queues with vacations are seen by au- thors like Levy and Yechailai [12], Doshi [13], Keilson and Servi [14]. A two stage **batch** **arrival** queuing system where customers receive a **batch** service in the first and individual service in the second stage was studied by Doshi [15] in the past. In recent years, extensive amount of work has been done on **batch** **arrival** queues with va- cations and breakdowns. We mention a few recent papers by Kumar and Arumuganathan [16], Choudhury, Tadj and Paul [17], Maraghi, Madan and Darby-Dowman [18], Khalaf, Madan and Lukas [19].

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Although the bounds illustrated in Figures 2-5 are seemingly accurate, the bounds degrade with the level of correlations within the arrivals. This trend can be particularly noticed for 1-order vs. 2-order autoregressive **processes** (see Figure 2.3 vs. 2.3); the same could be observed by reducing the scale of the x-axis in Figures 2.3 and 2.3. One explanation is that on a logarithmic y-axis the simulations throughout are seemingly convex, i.e., the probabilities in an initial phase decay faster than asymptotically (this behavior has been in fact convincingly shown to hold for bursty flows in [3]). In contrast, as the martingale-envelope is based on an exponential transform, it can only render bounds of the form of the (generalized) exponential distribution (i.e., P rob ≤ κe −θx ), whence the straight lines in the plots. In other words, the longer the

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I n this paper a Finite Waiting Space **Markovian** Single – Server Queueing Model with discouraged arrivals, retention of reneged customers and controllable **arrival** rates is considered. The steady state solutions and the system characteristics are derived for this model. The measures of effectiveness of the queueing model are also obtained. The analytical results are numerically illustrated.

Hadidi, 1985; Hadidi, 1981; Conolly and Choo, 1979 have analysed the M/M/1 queue where the service time and the preceding inter-**arrival** time have a bivariate exponential density with a positive correlation. Queueing systems with the number of channels and **arrival** rate depending on the system size are well known. In the human server production system the speed of the server and the rate in which jobs arrive at the system depend on the amount of work present (Bekker et al., 2004; Asmussen, 2003). In packet-switched communication system, feedback information on the buffer state provides the basis for the transmission control protocol to carefully regulate the transmission rate of internet flows.

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Abstract- **Batch** **arrival** feedback queue with additional multi optional service and multiple vacation is considered. All the arriving customers demand first essential service and only some of them demand second optional service. After the completion of second service, customer may feedback to the tail of original queue to repeat the service until it is successful or may depart forever from the system. If there is no customer in the queue the server goes on multiple vacation. Service times are gernerally distributed and vacation time is exponentially distributed. The time dependent probability generating functions has been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Mean queue length and mean waiting time are computed.

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The M/G/1 classic queueing system were extended by many authors in last two decades. The systems with server’s va- cation are important models that extend the M/G/1 queueing system. Also another conditions such as admissibility re- stricted may occure in systems. From this motivation, in this system I consider a single server queue with **batch** **arrival** Poisson input. There is a restricted admissibility of arriving batches in which not all batches are allowed to join the sys- tem at all times. At each service completion epoch, the server may apt to take a vacation with probability θ or else with probability 1 − θ may continue to be available in the system for the next service. The vacation period of the server has two heterogenous phases. Phase one is compulsory, and phase two follows the phase one vacation in such a way that the server may take phase two with probability p or may return back to the system with probability 1 − p. The vacation times are assumed to be general. All stochastic **processes** involved in this system (service and vacation times) are inde- pendent of each other. We derive the PGF’s of the system and by using them the informance measures are obtained. Some numerical approaches are examined the validity of results.

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This paper studies the stability problem of networked control system by modeling the random time delays and packet dropouts as two Markov chains. Active time varying sampling period strategy is proposed to make sure the time delay between sensor and actuator is shorter than one sampling interval, which conducts the closed- loop NCS as a **Markovian** jump linear system. Sufficient conditions of stochastic stability for the jump linear systems are given in terms of a set of LMIs. To solve the LMIs for obtaining feedback gains, the “gridding approach” is adopted to guarantee the LMIs set for the jump linear system with finite jump modes. It is shown the state feedback gain is mode-dependent. Numerical examples illustrate the effectiveness of the proposed strategy for the stochastic stabilizing controller over NCS.

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Nowadays, Tandem queueing model has drawn a good deal of attention for many researchers be- cause of it’s practical applications in real world. Hunt[8] studied a finite server tandem queueing sys- tem and discussed the notion of approximate decom- position. Burke[4] has considered a queueing system with the poisson **arrival** flow and exponential dis- tributed service times, he showed that the output process also follow poisson with same parameter as

Abstract. In this paper, we derive the perturbation and post **Markovian** perturbation to non-**Markovian** equation of motion(NMEM) that correspond to coherent and quad- rature non-**Markovian** stochastic Schr¨ odinger equations (SSE). In that case, we derive two perturbation approaches for zero and first orders to the coherent and quadrature NMEM. In order to explain both approaches, we apply two examples of non-**Markovian**. Keywords: perturbation method, post **Markovian** perturbation method, non-**Markovian** equation of motion, probability operator, functional operator.

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Single server queueing system in literature assumes that the server provide one type of general service with same mean rate to all the customers. But in real life situations there could be variation in mean service rate due to many reasons. Baurah et al. (2014) studied a **batch** **arrival** single server queue with server providing general service in two fluctuating modes. Madan (2014) investigated a **batch** **arrival** queue with general service in three fluctuating modes, balking, randomized breakdowns and a stand-by server during breakdown periods.

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The mild and clean conditions for these aziridinations with few side products enabled them to be combined with further ring opening or formal cycloadditions. A variety of alkenes were aziridinated and the in situ aziridines directly reacted with various nucleophiles, in telescoped flow **processes**. This telescoped flow process enabled the synthesis of a range of nitrogen containing molecules (22 examples) directly from the alkene, without having to isolate and purify the intermediate aziridine. This is very useful in cases where the aziridines formed are highly reactive and difficult to isolate, such as 2-aryl N- o Ns aziridines, or are

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