A SPECTRAL APPROACH TO COMPUTE THE MEAN PERFORMANCE MEASURES OF THE QUEUE WITH LOW-ORDER BMAP INPUT

This paper targets engineers and practitioners who want a simple procedure to compute the mean performance measures of the Batch Markovian Arrival process (BMAP/G/1) queueing system when the parameter matrices order is very low. We develop a set of system equations and derive the vector generating function of the queue length. Starting from the generating function, we propose a spectral approach that can be understandable to those who have basic knowledge of M/G/1 queues and eigenvalue algebra.


Introduction
The purpose of this paper is to present, to practitioners and engineers, a simple approach to compute the mean performance measures of the queueing system into which customers arrive according to a typical non-renewal arrival process called the BMAP (Batch Markovian Arrival Process).The BMAP, with parameter matrices D k , (k ≥ 0), is a Markov process {X(t), J(t)} with infinitesimal generator where X(t) represents the number of arrivals during (0, t) and J(t) represents the phase of the underlying Markov chain (UMC) at t. D 0 has negative diagonal elements and non-negative off-diagonal elements, {D k , (k ≥ 1)} are non-negative arrival rate matrices, and D = ∞ k=0 D k is an irreducible infinitesimal generator of the UMC.Special BMAP cases include the MAP (Markovian Arrival process); the MMPP (Markov Modulated Poisson process); the IPP (Interrupted Poisson process); the phase-type renewal process; the Poisson process; and various superpositions of these.For the definition and more comprehensive treatment of the BMAP, readers are advised to see Lucantoni [7], Lucantoni et al. [8], and Latouche and Ramaswami [6].
Well-established theories and computational algorithms exist concerning BMAP/G/1 queues with arbitrary order of parameter matrices.However, there are many cases where real data is fit to one of the BMAP schemes, resulting in order of matrices that are very low for practical purposes (Heffes and Lucantoni [5], Yousef and Schormans [17]).Despite these low-order parameter matrices, it is a formidable task for practitioners, who have completed Gross and Harris [4], Cooper [2], or Takagi [16], to understand published theories and write computer code even for the simplest MAP/G/1 queues.This has motivated our research.
Classical theory on BMAP/G/1 is based on the matrix-analytic method pioneered by Neuts [9,10].The BMAP was first introduced as a versatile Markovian point process in Neuts [11].The BMAP/G/1 queue was first analyzed by Ramaswami as a N/G/1 queue [15].Computational algorithms for BMAP/G/1 queues are provided in Lucantoni [7].Another approach to the BMAP/G/1 queue is spectral analysis based on eigenvalue algebra.Advanced readers are advised to reference Gail, Hantler and Taylor [3], Nishimura [12], Nishimura and Jiang [13], and Nishimura and Sato [14] (and references therein).Although our theoretical background is very similar to those cited above, our approach is more practical because our spectral approach is intended for field practitioners and provides step-by-step procedures to follow.
We initially derive the system equations for queue length by adopting the supplementary variable technique.We then use the eigenvalues and eigenvectors of the matrix generating function −D(z) = − ∞ n=0 z n D n to derive the vector generating function for queue length and mean queue length.

System equations
Let us define the following notations and probabilities: N (t): queue length (number of customers in the system including the one in service) at nD n e: mean arrival rate, ρ = λE(S): traffic intensity, (D n ) i,j dt: probability that during (t, t + dt] an arrival of group size n occurs and the phase of the UMC changes from i to j just after the arrival, under the condition that UMC was in phase i at t, q i = P r[server is idle at t, J(t Denoting (D n ) i,j as the (i, j) element of the matrix D n , it is easy to derive the following infinitesimal system equation for q(t): where 1 + (D 0 ) i,i dt is the probability that no changes occur during (t, t + dt] in both the queue length and the phase of the UMC.Also we get Taking t → ∞, and using q = (q 1 , . . ., q m ) and p n (x) = (p n,1 (x), . . ., p n,m (x)), we get the following vector system equations: Equation (2.1) represents the balance of in-flow and out-flow in UMC phases with zero customers in the system.Equation (2.2) represents the balances in UMC phases when there are n customers in the system.

Vector generating function of the queue length
Let us define the following vector and matrix generating functions, We multiply equation (2.2) by z n , sum over n = 1, 2, . .., and use equation (2.1) to get: x)e −θx dx.We take the LT of both sides of equation (2.3) to get In order for p * (z, θ) to be complete, p(z, 0) must be determined.This can be accomplished by making the left-hand side of equation (2.4) zero.For this purpose, let α 1 (z), α 2 (z), . .., α m (z) be the eigenvalues of the matrix GF −D(z) = − ∞ n=0 D n z n , and ξ i (z) be the right eigenvector of α i (z).If we use the eigenvalue α i (z) in θ of equation (2.4) and postmultiply both sides by its right eigenvector ξ i (z), we get Because α i (z) and ξ i (z) are related by −D(z)ξ i (z) = α i (z)ξ i (z), the left-hand side of equation (2.5) vanishes and yields: The above identity should hold for any eigenvalue of −D(z) and its right eigenvector.Thus we have where φ i (z) = αi(z)S * (αi(z)) S * (αi(z))−z .For further analyses, we make the following two assumptions: For other analyses of various queueing systems under these assumptions, readers are advised to see Nishimura [12], Nishimura and Jiang [13], and Nishimura and Sato [14].

Obtaining the unknown vector
The vector PGF from equation (2.16) is complete only after the unknown vector q = (q 1 , . . ., q m ) is obtained.Note that q i is the joint probability that the server is idle, and the UMC is in phase i at an arbitrary point of time in steady-state.The vector q can be obtained by solving the m simultaneous equations that result from using the zeros z * 1 , . . ., z * m , (|z * i | ≤ 1) of the denominator {S * (α i (z)) − z, i = 1, 2, . . ., m} in the numerator.
Theorem 2.1: There exists, for some i, an eigenvalue α i (z) that satisfies

. , α m (1).
Let α be an eigenvalue and ξ be its right eigenvector.We then have −Dξ = αξ.The row sum of the matrix −D is zero since it is the infinitesimal generator of the UMC.Thus α = 0 and ξ = e satisfies −Dξ = αξ, which completes the proof.

. , m).
Proof: Let A n (t) be the probability matrix that n customers arrive during t in a BMAP.Let A(z, t) = ∞ n=0 A n (t)z n .We then have d dt A(z, t) = A(z, t)D(z) (Lucantoni et al. [8]), the solution to which is A(z, t) = e D(z)t .Expressing e D(z)t in its Jordan form and taking t → ∞ shows that D(z) is stable on {z : |z| ≤ 1}.From Theorem 2.3, with a 1 (1) = 0, real parts of the eigenvalues of D(z) are negative on {z : |z| ≤ 1}.From Theorem 2.4, eigenvalues α 2 (z), . . ., α m (z) of −D(z) have positive real parts on |z| ≤ 1.
From Theorem 2.6, we see that the α i (z) that satisfies α i (1) = 0 is unique.

Theorem 2.7:
The denominator S * (α j (z)) − z of γ j (z) in equation (2.16) has, for each j (j = 2, 3, . . ., m), exactly one zero z * i within the unit circle.Proof: Let the closed contour C be C = {z : |z| = 1}.Let f (z) = −z and g(z) = S * (α j (z)).Denote α j (z) as α j (z) = a j + b j i, where a j and b j are real and imaginary parts.We note from Theorem 2.6 that a j > 0 on C for j = 2, 3, . . ., m.Now, we have |f (z)| = | − z| = 1 on C, and From the well-known Rouche's theorem, f (z) and f (z) + g(z) have the same number of zeros within the unit circle.It is obvious that f (z) = −z has exactly one zero within the unit circle.Thus, f (z) + g(z) = S * (α j (z)) − z has exactly one zero within the unit circle.
The numerator of equation (2.16) should vanish at the zeros z * 2 , z * 3 , . . ., z * m .We derive (m − 1) equations that will be used to determine q.One remaining equation can be obtained from the following theorem.
We can obtain the m equations to determine q = (q 1 , q 2 , . . ., q m ) completely.

Mean Queue Length
The mean queue length can be obtained from equation (2.16) as, where and, for i = 2, 3, . . ., m, .
L q can be obtained from From the Little's law, we respectively have mean sojourn time W and mean waiting time W q as

A numerical example
We consider a BMAP/G/1 queue with parameter matrices D 0 = −2. .
We compared the above mean performance measures with the ones we got from the algorithms of Lucantoni et al. [8], and confirmed that they are equal.

Summary and Discussion
In this paper, we have presented a simple spectral method to calculate mean performance measures of the queues with a non-renewal inputs.Our motivation was based on the fact that most practitioners who have completed basic queueing text books find it very difficult to understand the theories of the matrix-analytic method and program the published algorithms.
Our method is restrictive in that: (i) it is based upon the assumption that the eigenvalues of the matrix generating function D(z) are all distinct, and (ii) the eigenvalues of the matrix GF D(z) are easy to find.
For most practical problems with low-order parameter matrices, restriction (i) may not pose significant problems.But one may have a problem in finding all the eigenvalues as a function of z.In this case, commercially available mathematical software packages may simplify this task.

.. 2 . 3 : 2 . 4 : 2 . 5 : 2 . 6 :
From πD = 0 and πe = 1, the above equation is reduced to Then π d dz D(z)e z=1 = π ∞ n=0 nD n e = λ completes the proof.The result in (b) is a consequence of part (a).Theorem(Neuts [9]:pp 40) If A is an irreducible matrix with negative diagonal elements and non-negative off-diagonal elements, such that Ae ≤ 0, then A has a simple, non-positive eigenvalue −a(a ≥ 0) such that for all other eigenvalues a j of A, we have Re(a j ) < −a.Theorem We have Re(a j (1)) > 0, (j = 2, 3, . . ., m).Proof: D(1) = D(z)| z=1 = D is the infinitesimal generator of the UMC.Thus the diagonal elements are negative, the off-diagonal elements are nonnegative, and De = 0. Thus D is eligible for being A (from Theorem 2.3).The eigenvalue α 1 (1) = 0 of −D is also an eigenvalue of D. Thus all other eigenvalues of D have negative real parts.This means that all other eigenvalues of −D have positive real parts; thereby, completing the proof.Theorem (Bellman[1]) A matrix A is called stable if the solution of dx dt = Ax, x(0) = c approaches zero as t → ∞, regardless of the value of c.A necessary and sufficient condition for the stability of A is that all the eigenvalues of A have negative real parts.TheoremFor z on {z : |z| ≤ 1}, we have Re(a j (z)) > 0, (j = 2, . .

1 .
We assume that the service time follows an Erlang distributiuon of order 2 with the LT S * (θ) = 10 θ+10 2