USING FACTORIZATION IN ANALYZING D-BMAP/G/1 QUEUES

The discrete-time batch Markovian arrival process (D-BMAP) was first defined in [2]. The D-BMAP can represent a variety of arrival processes which include, as special cases, the Bernoulli arrival process, the Markov-modulated Bernoulli process (MMBP), the discrete-time Markovian arrival process (D-MAP), and their superpositions. It is the discrete-time version of the versatile Markovian point process introduced by Neuts [28], the N-process of Ramaswami [31], and the batch Markovian arrival process of Lucantoni [25, 26]. The objective of this paper is to demonstrate how one can apply the factorization property to the derivation of the queue-length distributions of the D-BMAP/G/1 queues with complex operational behavior during the idle period. To demonstrate how this new approach works, we are going to analyze the D-BMAP/G/1 queueing system under a double threshold policy and a setup time, which becomes the basic model for many production systems. The approach in this paper is simpler than the conventional matrix analytic method (MAM) and the supplementary variable technique. The MAM was pioneered by Neuts [29]. It starts with the analysis of the imbedded Markov renewal process at departure epochs. This method is cumbersome, especially in a system with a high degree of behavioral complexities during the idle period, in that it involves manipulating the vast amount of matrices without knowing the practical meaning of the resulting matrices. Works based on MAM are many. Blondia and Casals [2] modeled a digital video communication system by D-BMAP. Hashida et al. [7] analyzed the system with switched batch Bernoulli process (SBBP) with and without priorities. Ishizaki et al. [10] analyzed the SBBP/G/1 system in which the staying time of the underlying


Introduction
The discrete-time batch Markovian arrival process (D-BMAP) was first defined in [2].The D-BMAP can represent a variety of arrival processes which include, as special cases, the Bernoulli arrival process, the Markov-modulated Bernoulli process (MMBP), the discrete-time Markovian arrival process (D-MAP), and their superpositions.It is the discrete-time version of the versatile Markovian point process introduced by Neuts [28], the N-process of Ramaswami [31], and the batch Markovian arrival process of Lucantoni [25,26].
The objective of this paper is to demonstrate how one can apply the factorization property to the derivation of the queue-length distributions of the D-BMAP/G/1 queues with complex operational behavior during the idle period.To demonstrate how this new approach works, we are going to analyze the D-BMAP/G/1 queueing system under a double threshold policy and a setup time, which becomes the basic model for many production systems.The approach in this paper is simpler than the conventional matrix analytic method (MAM) and the supplementary variable technique.
The MAM was pioneered by Neuts [29].It starts with the analysis of the imbedded Markov renewal process at departure epochs.This method is cumbersome, especially in a system with a high degree of behavioral complexities during the idle period, in that it involves manipulating the vast amount of matrices without knowing the practical meaning of the resulting matrices.Works based on MAM are many.Blondia and Casals [2] modeled a digital video communication system by D-BMAP.Hashida et al. [7] analyzed the system with switched batch Bernoulli process (SBBP) with and without priorities.Ishizaki et al. [10] analyzed the SBBP/G/1 system in which the staying time of the underlying

The factorization property
Chang et al. [4] proved that for the D-BMAP/G/1 queues with generalized vacations (based on late-arrival model (see Takagi [35])), the following factorization properties hold: where In (2.1), Y(z) is the vector generating function (GF) of the queue length at an arbitrary slot boundary and X(z) is the vector GF of the queue length just after an arbitrary departure.p idle (z) is the vector GF of the queue length at an arbitrary slot boundary during an idle period.In (2.2), D(z) = ∞ n=0 D n z n , where D n is the matrix of the arrival probabilities for a group of size n during a slot.λ = π ∞ n=1 nD n e is then the mean arrival rate per slot where π is the stationary vector of the UMC that satisfies ] k is the matrix GF of the number of customers that arrive during a service time in which s k is the probability that a service time is of length k slots.
From (2.1), we note that the following relationship holds between X(z) and Y(z): which confirms Kim et al. [15].Equation (2.1) implies that if one wants to derive the GFs Y(z) and X(z), all they need to do is derive the vector GF p idle (z) of the queue length at an arbitrary slot boundary during an idle period.

The system
In this paper, we deal with the queueing system with the following specifications (Figure 3.1).
(1) Customers arrive according to the D-BMAP in which the UMC is governed by parameter matrices {D 0 ,D 1 ,D 2 ,...}.We assume that the UMC has m phases.(2) If there are no customers to serve (point 1 in Figure 3.1), the server waits until the queue length reaches or exceeds the first threshold α (buildup period).(3) At the end of the buildup period, the server starts a setup period which takes a random time H with h k = Pr(H = K).At the end of the setup period, (i) if the queue length is less than the second threshold N, the server waits until queue length reaches or exceeds N (standby period); or (ii) if the queue length is greater than or equal to N, the server begins to serve the customers (busy period).(4) The service time S is a random variable with s k = Pr(S = K).
(5) The service times and setup times are independent of the arrival process and the phases of the UMC.The objective of this study is to analyze the above queueing system by applying the factorization property of the D-BMAP/G/1 queue with generalized vacations directly to 122 Using factorization in analyzing D-BMAP/G/1 queues the vector generating functions of the queue lengths.We also derive the mean queue length.
This queueing system can be used to model the production system in which the setup cost is extremely high.Lee and Park [20] showed that the double threshold (α,N)-policy is better than the single threshold N-policy when the setup cost is extremely high compared to the work-in-process (WIP) holding cost.We note that Lee et al. [19] applied the factorization to the analysis of WIP of a production system.
If α = N, we have the usual N-policy queue with a setup.N-policy queue was first studied by Yadin and Naor [37].For other works on N-policy queues, see, for example, [8,9,14,18].

The analysis
From (2.1), we only need to derive the vector GF p idle (z) of the queue length at an arbitrary slot boundary during an idle period.
We define p bu , p su , and p sb as the probabilities that the server is in the buildup period, setup period, and standby period, respectively, under the condition that the server is idle.If we define p bu (z), p su (z), and p sb (z) as the conditional vector GF of the queue length at an arbitrary point within the buildup, setup, and standby periods, respectively, we get To obtain the probabilities p bu , p su , and p sb , we need the mean lengths E(I) of an idle period, E(T bu ) of a buildup period, and E(T sb ) of a standby period.In the sequel, we will use (F) i j to denote the (i, j)-element of the matrix F.
We define the probabilities (Ψ bu k ) i j and (Ψ sb k ) i j as follows: (a) (Ψ bu k ) i j denotes Pr (the buildup process ever visits level k and the phase of UMC is j just after the visit | UMC phase is i at 1 of Figure 3.1); (b) (Ψ sb k ) i j denotes Pr (the standby process ever visits level k and the phase of UMC is j just after the visit | UMC phase is i at 1 of Figure 3.1).Let κ = (κ 1 ,κ 2 ,...,κ m ) be the probability vector of the UMC phase at 1 .Noting that the (i, j)-element of the matrix (I − D 0 ) −1 is the mean time the UMC stays in phase j until the next arrival given the current phase is in i, we have Then, we get ) ) Noting that the jth element of the vector is the probability that the queue length is k and the UMC phase is j under the condition that the server is in the buildup period, we get which yields, from (4.4a), Analogously, we get To obtain p su (z), let H − α (z) be the matrix GF of the queue length at the start of the setup period (point 2 of Figure 3.1).Then, we have where H e (z) is the matrix GF of the number of customers that arrive during the elapsed time, which is given by in which (4.9) Now, we need a scheme to compute the probabilities , and κ.To this end, we observe that the behavior of the queueing process during the buildup period is exactly the same as that of the usual D-BMAP/G/1 system with simple N-policy (i.e., without setup time) where N is replaced by α.  n ) i j be the probability that the idle period process of the D-BMAP/G/1/N-policy queue ever visits level n and the UMC phase just after the visit is j given that the UMC phase is i at a of Figure 4.1.Then, Proof.D * 0 = I is obvious.Noting that (I − D 0 ) −1 D n is the phase transition probability by the arrival of a group of size n, the proof is complete by conditioning on the level k visited prior to level n.Now, returning to our system, we get the following theorem without proof.Theorem 4.2.
) is in order.We first note that Ψ sb k depends on both the queue length at the start of the setup period and the number of customers that arrive during the setup time.Noting that the queue length at the start of the setup Ho Woo Lee et al. 125 time is exactly equal to the queue length in the simple D-BMAP/G/1/N-policy queue where N is replaced by α, we present the following lemma.Lemma 4.3 (BMAP/G/1/N-policy queue).Let (Q N n ) i j (n ≥ N) be the probability that the queue length is n and the UMC phase is j at b of Figure 4.1 given that the UMC phase is in i at a .Then, there exists a recursion ) Proof.Equation (4.12a) can be obtained by conditioning on the size of the first arrival group; (4.12b) can be obtained by mathematical induction.
We define the following notations: (i) (H − k(α) ) i j denotes the joint probability that the queue length is k and the UMC phase is j at 2 of Figure 3.1 given that the UMC phase is i at 1 , (ii) (H + k(α) ) i j denotes the joint probability that the queue length is k and the UMC phase is j at 3 of Figure 3.1 given that the UMC phase is i at 1 , (iii) (H k ) i j denotes the probability that k customers arrive during the setup time and the UMC phase is j at 3 under the condition that the UMC phase is i at 2 .We define the matrix GFs of the above probability matrices as follows: Then, we have the following theorem.
Theorem 4.4.There exist 126 Using factorization in analyzing D-BMAP/G/1 queues Proof.Equations (4.14a) and (4.14b) can be obtained by using α in place of N in (4.12a) and (4.12b).Equations (4.14c) and (4.14d) are obvious.Equation (4.14e) can be obtained first by conditioning on the queue length at the end of the setup period and then by applying (4.10).
So far, we have obtained all the quantities that we need in (4.9) except κ, which is the stationary phase probability vector at 1 .Let K be the phase transition probability matrix between 1 and 5 .Then, κ can be computed from (see Lucantoni [25]) where K(z) is the matrix GF of the number of customers that are served between 1 and 5 .To obtain K(z), we need the information concerning the queue length at 4 .Let (Q (α,N) k ) i j be the probability that the queue length is k and the UMC is in phase j at 4 under the condition that the UMC is in phase i at 1 (k ≥ N).We define the matrix GF . Then, we have the following.
Proof.Equation (4.17a) can be obtained by conditioning on the queue length at the end of the setup period.Multiplying (4.17a) by z n and summing over n , we get From the identity ∞ n=N− j Q N− j n z n = Q N− j (z), and using (4.12b) and (4.14e), we get the second term of the last equality as Ho Woo Lee et al. 127 which proves (4.17b).Now, we have ) In (4.20a), (4.20b), G(z) is the matrix GF of the number of customers that are served during a fundamental period (see Neuts [29]) which is given by and G is the phase transition matrix during the fundamental period, which can be obtained as follows: . Now, by computing κ from (4.15) with K obtained in (4.20b), we can determine all the quantities that we need for complete p idle (z) in (4.9).

Figure 4 .
1 shows a sample path for the D-BMAP/G/1 queue with the simple N-policy.