© Hindawi Publishing Corp. FLUID QUEUES DRIVEN BY A DISCOURAGED ARRIVALS QUEUE

We consider a fluid queue driven by a discouraged arrivals queue and obtain explicit expressions for the stationary distribution function of the buffer content in terms of confluent hypergeometric functions. We compare it with a fluid queue driven by an infinite server queue. Numerical results are presented to compare the behaviour of the buffer content distributions for both these models. 2000 Mathematics Subject Classification: 60K25. 1. Introduction. Stochastic fluid flow models are increasingly used in the performance analysis of communication and manufacturing models. Recent measurements have revealed that in high-speed telecommunication networks, like the ATM-based broadband ISDN, traffic conditions exhibit long-range dependence and burstiness over a wide range of time scales. Fluid models characterize such traffic as a continuous stream with a parameterized flow rate. Fluid queue models, where the fluid rates are controlled by state-dependent rates, have been studied in the literature. van Doorn and Scheinhardt [3] analyse the content of the buffer which receives and releases fluid flows at rates which are determined by the state of an infinite birth-death process evolving in the background. Lam and Lee [7] investigate a fluid flow model with linear adaptive service rates. Lenin and Parthasarathy [9] provide closed form expressions for the eigenvalues and eigenvectors for fluid queues driven by an M/M/1/N queue. Resnick and Samorodnitsky [12] have obtained the steadystate distribution of the buffer content for M/G/∞ input fluid queues using large deviation approach. In this paper, we obtain explicit expressions for the stationary distribution function of the buffer content for fluid processes driven by two distinct queueing models, namely, discouraged arrivals queue and infinite server queue, respectively. Both these models have the same steady-state probabilities. We show that the buffer content distributions of fluid queues modulated by the two models vary considerably as depicted in the graph. The discouraged arrivals single-server queueing system is useful to model a computing facility that is solely dedicated to batch-job processing (see [11]). The well-known infinite server queue is often used to analyze open loop statistical multiplexing of data connections on an ATM network (see [6]).


Introduction.
Stochastic fluid flow models are increasingly used in the performance analysis of communication and manufacturing models.Recent measurements have revealed that in high-speed telecommunication networks, like the ATM-based broadband ISDN, traffic conditions exhibit long-range dependence and burstiness over a wide range of time scales.Fluid models characterize such traffic as a continuous stream with a parameterized flow rate.
Fluid queue models, where the fluid rates are controlled by state-dependent rates, have been studied in the literature.van Doorn and Scheinhardt [3] analyse the content of the buffer which receives and releases fluid flows at rates which are determined by the state of an infinite birth-death process evolving in the background.Lam and Lee [7] investigate a fluid flow model with linear adaptive service rates.Lenin and Parthasarathy [9] provide closed form expressions for the eigenvalues and eigenvectors for fluid queues driven by an M/M/1/N queue.Resnick and Samorodnitsky [12] have obtained the steadystate distribution of the buffer content for M/G/∞ input fluid queues using large deviation approach.
In this paper, we obtain explicit expressions for the stationary distribution function of the buffer content for fluid processes driven by two distinct queueing models, namely, discouraged arrivals queue and infinite server queue, respectively.Both these models have the same steady-state probabilities.We show that the buffer content distributions of fluid queues modulated by the two models vary considerably as depicted in the graph.The discouraged arrivals single-server queueing system is useful to model a computing facility that is solely dedicated to batch-job processing (see [11]).The well-known infinite server queue is often used to analyze open loop statistical multiplexing of data connections on an ATM network (see [6]).

Confluent hypergeometric function.
We obtain explicit expressions for the buffer content distributions of the fluid queues driven by discouraged arrivals queue and infinite server queue by employing well-known identities of confluent hypergeometric function.Some of the identities are presented in this section.
The confluent hypergeometric function, also referred to as Kummer function, is denoted by 1 F 1 (a; c; z) and is defined by for z ∈ C, parameters a, c ∈ C (c a nonnegative integer), with (α) n , known as Pochhammer symbol, defined as Observe that 1 F 1 (0; c; z) = 1 and 1 F 1 (a; a; z) = e z .The confluent hypergeometric function satisfies the relations (see [1]) (2.4) The following identities are from [2]: ) (2.7)

Model description.
Consider a fluid model driven by a single server queueing process with state-dependent arrival and service rates.It consists of an infinitely large buffer in which the fluid flow is regulated by the state of the background queueing process.Denote the background queueing process by ᐄ := {X(t), t ≥ 0} taking values in the state space of nonnegative integers, where X(t) denotes the state of the process at time t.Let λ n and µ n denote the mean arrival and service rates, respectively, when there are n units in the system.
The background queueing process modulates the fluid model in such a way that during the busy periods of the server, the fluid accumulates in an infinite capacity buffer at a constant rate r > 0. The buffer depletes the fluid during the idle periods of the server at a constant rate r 0 < 0 as long as the buffer is nonempty.We denote by C(t) the content of the buffer at time t.Clearly, the 2dimensional process {(X(t), C(t)), t ≥ 0} constitutes a Markov process, and it possesses a unique stationary distribution under a suitable stability condition.
The stationary state probabilities p i , i ∈ , of the background process are given by where .., and π 0 = 1 are called the potential coefficients.To ensure the stability of the process {(X(t), C(t)), t ≥ 0}, we assume the mean aggregate input rate to be negative, that is, Letting the Kolmogorov forward equations for the Markov process {X(t), C(t)} are given by + µ n+1 F n+1 (t, x), n ∈ \{0}, t,x ≥ 0 (3.4) (see [3]).Assume that the process is in equilibrium so that ∂F n (t, x)/∂t ≡ 0 and in that case lim t→∞ F n (t, x) ≡ F n (x).Hence, the above system reduces to a system of ordinary differential equations When the net input rate of fluid flow into the buffer is positive, the buffer content increases and the buffer cannot stay empty.It follows that the solution to (3.5) must satisfy the boundary conditions But F 0 (0) is nonzero and is determined later.Also, We study two fluid models driven by state-dependent queues with arrival and service rates given by ) For the process to be stable, from (3.2), (r 0 − r )+ r e ρ < 0 where ρ denotes the ratio λ/µ.Both the queueing models under consideration have the same steady-state probabilities given by p n = (ρ n /n!)e −ρ .From [13], the stationary probability for the fluid queue to be empty is given by for both these models.
Our task is to solve the system of (3.5) with rates suggested by (3.8) and (3.9) subject to conditions (3.6) and (3.7).The stationary buffer content distribution can then be obtained.
In this sequel, let Fn (s) denote the Laplace transform of the function F n (x).

Discouraged arrivals queue.
In this section, we consider a fluid queue driven by a state-dependent queueing model with rates given by (3.8) and obtain an explicit expression for the quantity F n (x) using well-known identities of confluent hypergeometric functions.As suggested by the birth and death rates, it is seen that the arrivals decrease as the queue length increases and hence the name discouraged arrivals queue.The governing system of forward Kolmogrov equations for this model is Laplace transform yields We obtain the solution of the above system of equations in terms of confluent hypergeometric function.Defining it is observed that (4.3) reduces to We identify that the term ĝn (s) satisfies (2.3) with a = n+2, c = λr s/(r s + µ) 2 + n + 2, and z = −λµ/(r s + µ) 2 .Thus, we can deduce from (4.5) and (2.3) that and hence Fn (s) In order that Fn (s) satisfies (4.2), we redefine where Now, we invert (4.9) by expanding the function as (4.11) Laplace inversion yields where φ * (j) (x) denotes the j-fold convolution of φ(x), where δ(x) is the Dirac delta function.We now verify the boundary condition (3.7).Using the fact that 1 F 1 (a,a,z) = e z , observe that 1 F 1 (2,λr s/(r s + µ) 2 + 2, −λµ/(r s + µ) 2 ) and hence φ(s) tends to e −ρ as s → 0. Hence we have (from (3.10)).

Infinite server queue.
In this section, we consider a fluid queue driven by an infinite server queue and obtain an explicit expression for G n (x), thereby highlighting the variation in their expressions, although both the underlying queueing models have the same steady-state probabilities.For the sake of clarity in notation, we use G n (x) in place of F n (x).The forward Kolmogrov equations for this model are (5.1) Laplace transform yields Here, G 0 (0) = F 0 (0).Analysing as before, if then (5.3) reduces to (5.5) We observe that kn (s) satisfies the recurrence relation (2.3) with a = n + 1, c = r s/µ + n + 1, and z = −λ/µ.Thus we have By a similar argument as in the previous section in order to satisfy (5.2), we redefine where (5.9) Subject to the above definition, the verification of (5.2), being satisfied by Ĝ0 (s) and Ĝ1 (s), is done through certain algebra involving the application of identities (2.5) and (2.6) (see Appendix B).
To facilitate the Laplace inversion, we write Ĝn where (5.11) On inversion, we get (5.12) Hence, we have where Using 1 F 1 (a,a,z) = e z , we verify below the boundary condition (3.7) for the buffer content distribution as In this way, we analytically obtain closed form expressions for F n (x) and G n (x) for both the models as given by (4.12) and (5.13), respectively.Hence, we obtain the stationary distribution of the buffer content given by where F 0 (0) is given by (3.10).

Asymptotic analysis.
In this section, we discuss the large deviations calculation that gives the asymptotic straight line fit to the two models under consideration.Large buffers are obtained by having the birth-death process avoid zero more often than average.Suppose that for a time t the average occupancy of the state zero in the birth-death process is x, then the drift of the fluid buffer is on average r 0 x + r (1 − x) = r − x(r − r 0 ), which is positive.The probability that the occupancy of state zero is near x is obtained by Sanov's theorem.Let m(t) represent the fraction of time that the birth-death process is zero in [0,t].Then P m(t) ≈ x = exp − tI(x) , ( where and H(x) = {ν i : i ν i = 1, ν i ≥ 0, ν 0 = x}.Following standard arguments as sketched in Schwartz and Weiss [14, Section 2.4], we use a Lagrange multiplier to find the minimum in I(x) as follows.We write I(x) = inf ν∈H(x) i p i α i log α i , where α i = ν i /p i for i > 0 and α 0 = x/p 0 .We then look for extreme points of the function where the Lagrange multiplier K is chosen so that the condition i α i p i = 1 is satisfied.Setting the partial derivatives of the function with respect to α i equal to zero, for i > 0, we obtain that all α i , i ≥ 1, are equal, say α.Therefore Hence, we find that where x is the parameter to be determined.Recall that p 0 = e −ρ .Now to estimate the probability that the fluid buffer is above some level B, we estimate the probability that m(t) is near x for sufficient time t.Note that the fluid buffer fills at rate r −x(r −r 0 ) so that the time required is t = B/(r − x(r −r 0 )).Therefore, the probability that the buffer fills to B is approximately We can find x which minimise the quotient I(x)/(r − x(r − r 0 )).We numerically determine the value of x and this minimum is unique.For example,   when r = 1 = −r 0 , with λ = 1 and µ = 1.7, we find x = 0.44469, quotient = 0.22213441870537, that is, P (fluid > t) ≈ exp(−0.22213441870537t).Observe that for sufficiently small values of t, exp(−0.22213441870537t) turns out to be a straight line.Figure 6.1 depicts the behaviour of this function I(x)/(r −x(r −r 0 )) against x for r = 1 = −r 0 and the varying values of λ and µ where I(x) is given by (6.5).Table 6.1 gives the value of x at which the quotient attains minimum for the various parameter values.

Numerical illustrations.
In this section, we briefly discuss the method of numerically evaluating the stationary buffer content distribution for the two models under consideration.
The governing system of differential-difference equations given by (3.5) can be written in the matrix notation as where ..] T , R = diag{r 0 ,r ,r ,...}, and Q denotes the infinitesimal generator of the background birth and death process given by 2) The capacity of the background birth and death process is unrestricted in our theoretical study.However, for the purpose of numerical investigations, we truncate the size of the process by a finite quantity, say N. Hence R −1 Q T takes the form Mitra [10] have shown that R −1 Q T has exactly N + negative eigenvalues, N − − 1 positive eigenvalues, and one zero eigenvalue, where N + is the cardinality of the set S + ≡ {j ∈ : r j > 0} and N − is that of S − ≡ {j ∈ : r j < 0}.
Suppose that ξ j , j = 0, 1, 2,...,N, are the eigenvalues of the matrix and y = [y 0 ,y 1 ,...,y N ] T , z = [z 0 ,z 1 ,...,z N ] are the right and the left eigenvectors of the matrices R −1 Q T and Q T R −1 , respectively, corresponding to the eigenvalue ξ .Then, where the polynomials B j (s) are recursively defined as follows: and From the knowledge of the eigenvalues, left and right eigenvectors, the equilibrium distribution of the buffer occupancy is given by (see [8]) where The unknown F 0 (0) representing the distribution of the buffer occupancy when the buffer is empty and the background process is in state zero is obtained as Determination of eigenvalues.We determine the eigenvalues of R −1 Q T from its associated characteristic polynomial denoted by P(s): It can be written as Doing the operations: (1) row i = (row i) + (row i + 1) for i = 1, 2,...,N, (2) diminishing the second column by the first, the third column by the new second column, and so on in the above determinant, we get Thus zero is an eigenvalue of R −1 Q T .The above determinant is sign-symmetric and hence can be written as The other eigenvalues of R −1 Q T are determined from the associated real symmetric matrix of this reduced determinant P(s) by using the method of bisection suggested by Evans et al. [4] with suitable modifications.
Determination of eigenvectors.Let M(s) = ((a ij )) denote the matrix (sI − R −1 Q T ) where I is the identity matrix of order N + 1.In expression (7.5) of the eigenvector of the underlying matrices, the polynomials B j (s) play a major role.Since the system of equations given by (7.6) is an underdetermined system, at least one of the equations is redundant.If the kth equation is redundant, we may assume B k (s) = 1 and solve the rest of the equations.Fernando [5] provides a method to overcome the instability that arises because of this particular normalization.This is achieved by computing the diagonal entries of the matrix Ᏺ, which is obtained by elementwise reciprocation of the inverse of M(s) T based on LDU and UDL factorization of the tridiagonal matrix M(s).
We consider the LDU factorization of M(s).The diagonal elements d i (s) of D are given recursively as where s is the eigenvalue of the matrix R −1 Q T .Now, we consider the UDL factorization of M(s).The diagonal elements δ i (s) are given recursively as Then the diagonal elements η i (s) of the matrix Ᏺ are given by , for i = 2, 3,...,N + 1. (7.17) The following algorithm may be used for computing B j (s) with suitable normalization suggested by the algorithm.(3) Compute other B j (s) using To visualize the foregoing discussion, we plot the graphs of buffer content distribution for the two models by assuming certain values for the parameter λ and µ.The variation in the buffer content distribution is well brought out by evaluating them numerically.Figure 7.1 depicts the behaviour of the stationary buffer content distribution against the content of the buffer x for both the models with r 0 = −1, r = 1 and N truncated at 30.It is observed from the graph that the buffer content distribution decreases with the increase in λ and decrease in µ.Adding and subtracting the term r (r s/µ + 1) 1 F 1 (1; r s/µ + 1; −λ/µ) and using identity (2.6), we obtain Hence the verification.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

Table 6 .
1. Value of x at which I(x)/(1 − 2x) attains the minimum and the corresponding minimum for different values of λ and µ.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation