Policies When the Input Process Is a Nonnegative L ´evy Process

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Volume 2011, Article ID 916952,17pages doi:10.1155/2011/916952

Research Article

Control of Dams Using P

_λ,τ^M

Policies When the Input Process Is a Nonnegative L ´evy Process

Mohamed Abdel-Hameed

Department of Statistics, College of Business and Economics, United Arab Emirates University, Al-Ain 17555, UAE

Correspondence should be addressed to Mohamed Abdel-Hameed,[email protected] Received 28 April 2011; Accepted 17 July 2011

Academic Editor: Ho Lee

Copyrightq2011 Mohamed Abdel-Hameed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We considerP_λ,τ^Mpolicy of a dam in which the water input is an increasing L´evy process. The release rate of the water is changed from 0 toMand fromMto 0M >0at the moments when the water level upcrosses levelλand downcrosses levelτ τ < λ, respectively. We determine the potential of the dam content and compute the total discounted as well as the long-run average cost. We also find the stationary distribution of the dam content. Our results extend the results in the literature when the water input is assumed to be a Poisson process.

1. Introduction and Summary

Lam and Lou 1 consider the control of a finite dam where the water input is a Wiener process, usingP_λ,τ^M policies. In these policies, the water release rate is assumed to be zero until the water reaches level λ > 0, as soon as this happens the water is released at rate M > 0 until the water content reaches levelτ > 0, λ > τ. Abdel-Hameed and Nakhi2 discuss the optimal control of a finite dam usingP_λ,τ^M policies, using the total discounted as well as the long-run average costs. They consider the cases where the water input is a Wiener process and a geometric Brownian motion process. Lee and Ahn3consider the long-run average cost case when the water input is a compound Poisson process. Abdel-Hameed4 treats the case where the water input is a compound Poisson process with a positive drift. He obtains the total discounted cost as well as the long-run average cost. Bae et al.5consider theP_λ,0^M policy in assessing the workload of an M/G/1 queuing system. Bae et al.6consider the log-run average cost forP_λ,τ^M policy in a finite dam, when the input process is a compound Poisson process. In this paper, we consider theP_λ,τ^M policy for the more general case where

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the water input is assumed to be an increasing L´evy process. At any time, the release rate can be increased from 0 toMwith a starting cost K₁Mor decreased from Mto zero with a closing costK₂M. Moreover, for each unit of output, a rewardR is received. Furthermore, there is a penalty cost which accrues at a ratef, wheref is a bounded measurable function on the state space of the content process.

We will use the term “increasing” to mean “nondecreasing” throughout this paper.

InSection 2, we discuss the potentials of the processes of interest as well as the other results that are needed to compute the total discounted and long-run average costs. In Section 3, we obtain formulas for the cost functionals using the total discounted as well as the long-run average cost cases. InSection 4, we discuss the special cases where the water input is an increasing compound Poisson process as well as inverse Gaussian process.

2. Basic Results

The content process is best described by the bivariate processB Z, R, whereZ{Zt, t≥0}

andR{Rt, t≥0} describe the dam content and the release rate, respectively. We define the following sequence of stopping times:

T₀inf{t≥0 :Z_t≥λ}, T^∗₀inf

t≥T₀:Z_t≤τ , Tninf

t≥T_n−1:Zt≥λ

, T^∗ninf

t≥Tn:Zt≤τ

, n1,2, . . . .

2.1

The processBhas as its state space the pair of line segments

S 0, λ× {0}∪τ,∞× {M}. 2.2

LetI {It, t≥0}be an increasing L´evy process with drifta≥0. For eacht≥0, we let I_t^∗I_t−Mt. From the definition of theP_λ,τ^M policy, it follows that, for eacht∈0,T₀,Z_tI_t,

Zt

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

It, t∈ ^∞

n0

_∗ Tn,T_n1

,

I_t^∗, t∈ ^∞

n0

Tn,T^∗n

.

2.3

Furthermore,I^∗

Tn I_T

n, n0,1, . . .. It follows that the content processZis a delayed regenerative process with the regeneration points being theT^∗_n, n1,2, . . . .The penalty cost rate function is defined as follows:

fz, r

⎧⎨

⎩

gz z, r∈0, λ× {0},

g^∗z z, r∈τ,∞× {M}, 2.4 whereg:0, λ → Randg^∗:τ,∞:→ Rare bounded measurable functions.

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For any processY {Yt, t ≥ 0} with state spaceE, any Borel set A ⊂ E and any functional f, E_yf denotes the expectation of f conditional on Y₀ y, P_yA denotes the corresponding probability measure, and I_A is the indicator function of the set A.

Throughout, we let R −∞,∞, R 0,∞, N {1,2, . . .}, and N {0,1, . . .}. For x, y ∈ R, we define x∨y xmaxy and x∧y xminy. Throughout, we defineWλ inf{t ≥ 0 : I_t ≥ λ} and W_τ^∗ inf{t ≥ 0 : I_t^∗ ≤ τ}. For any x < λ and y > τ, let C^α_g0, x, λandC^α_g∗M, y, τ be the expected discounted penalty costs, during the intervals 0, Wλ and 0, W_τ^∗, respectively. Furthermore, let Cg0, x, λ and Cg^∗M, y, τ be the expected nondiscounted penalty costs during the same intervals. It follows that

C^α_g0, x, λ Ex

_W_λ

0

e^−αtgItdt, C^α_g∗

M, y, τ Ey

_W_τ^∗

0

e^−αtg^∗I_t^∗dt,

C_g0, x, λ E_x _W_λ

0

gItdt, C_g^∗

M, y, τ E_y

_W∗ τ

0

g^∗I_t^∗dt.

2.5

The functionals above, which we aim to evaluate, are basic ingredients in computing the total discounted and long-run average costs associated with theP_λ,τ^M policy as discussed in Section 3.

Leta≥0 andνbe the drift term and the L´evy measure of input processI, respectively, then, for allt≥0, x≥0, andα≥0 the Laplace transform ofI_tis of the form,

E_x e^−αI^t

e^−txφα. 2.6

The functionφαis known as the L´evy component and is given by φα αa

_∞

0

1−e^−αx

νdx, 2.7

whereνis a measure on0,∞satisfying _∞

0

x∧1νdx<∞, ν{0} 0. 2.8

Increasing L´evy processes include increasing compound Poisson processes, inverse Gaussian processes, gamma processes, and stable processes.

We assume that the expected value ofI₁is finite throughout this paper.

To evaluate the cost functionals and other parameters of the content process, we define the L´evy process killed atWλas follows:

X{It, t < W_λ}. 2.9

From Theorem 3.3.12 of Blumenthal and Getoor7, it follows that the processXis a strong Markov process.

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Definition 2.1. LetYbe a Markov process with a state spaceE. For eachα≥0, theα-potential ofYdenoted byU^α_Yis defined for any bounded measurable function onE and everyx∈E via1.8.9, p.41 of8

U^α_Yfx^def

E

fzU^α_Yx, dz Ex

_∞

0

e^−αtfYtdt. 2.10

Remark 2.2. Throughout, we denote theα-potential of the processI byU^α. Since the process Ihas stationary independent increments, it follows thatU^αx, dy U^α0, dy−x, for each xandyin the state space of the processIsatisfyingy≥x. We denoteU^α0, dy byU^αdy, throughout.

Since the processI is increasing and has stationary independent increments, it follows that

C_g^α0, x, λ U^αgx _λ

x

g y

U^α x, dy

_λ−x

0

g xy

U^α dy

, 2.11

Cg0, x, λ U⁰gx _λ

x

g y

U⁰ x, dy

_λ−x

0

g xy

U⁰ dy

. 2.12

The following lemma follows by takinggx 1 for allx∈0, λ in2.11and2.12, respectively.

Lemma 2.3. Forx≤λone has Ex

exp−αWλ

1−αU^αI_0,λ−x0 αU^αI_λ−x,∞0, 2.13

ExWλ U⁰I_0,λ−x0. 2.14

The following Lemma gives the Laplace transform ofI_W_λas well as the expected value ofIWλ.

Lemma 2.4. aForx < λandα≥0,

Ex

exp−αIWλ

exp−αx

1−φα

0,λ−xexp−αzU⁰dz

. 2.15

bForx < λ,

ExIWλ xE0I1E0Wλ−x. 2.16

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Proof ofa. Forx < λandα≥0, since the processI has stationary independent increments, we have

E_x

exp−αIWλ E₀

exp−αxI_W_λ−x

exp−αx

φα

λ−x,∞exp−αzU⁰dz

exp−αx

φα

0,∞exp−αzU⁰dz−

exp−αx

φα

1 φα−

exp−αx

1−φα

,

2.17

where the second equation follows from 8 of Alili and Kyprianou 9, and the fourth equation follows from the definition ofφα andU⁰.

Proof ofb. Forx < λ,

ExIWλ xE0IWλ−x

xlim

α→0

1−E₀

exp−αIWλ−x α

xlim

α→0

φα α

xφ0U⁰I_0,λ−x0 xE0I1U⁰I_0,λ−x0 xE₀I1E0W_λ−x,

2.18

where the first equation follows since the process I is a L´evy process, the third equation follows from2.15, the fourth equation follows becauseφ0 0, the fifth equation follows sinceφ0 E₀I1, and the last equation follows from2.14.

To deriveC^α_g∗M, y, τ, Cg^∗M, y, τ, Eyexp−αW_τ^∗, andE_yW_τ^∗, we define

X^∗ _∗

It, t < W_τ^∗

. 2.19

Clearly, the state space of the processX^∗ isτ,∞. From Theorem 3.3.12 of Blumenthal and Getoor7, it follows that the processX^∗is a strong Markov process.

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Throughout, we assume thatM ≥ a. Using Doob’s optional sampling theorem, the following is easy to see.

Lemma 2.5. Forx≥τ, Ex

exp−αW_τ^∗ exp

−x−τηα

, 2.20

whereηαis the solution of the integral equation Mηα αφ

ηα

. 2.21

The following Lemma gives, among other things, a formula for computing E_xW_τ^∗and condition under which this expectation is finite.

Lemma 2.6. aη0 0 if and only ifM−E₀I1>0.

bThe functionηαis a concave increasing function onR. cForx≥τ,

E_xW_τ^∗ x−τ

M−E0I1 if M−E₀I1>0, ∞ otherwise.

2.22

Proof ofa. From 2.20, it follows that ηα is an increasing function on R and lim_α_{→ ∞}ηα ∞. Letfx η⁻¹x, using2.21it follows that fx Mx−φx. Fur- thermore, η0is the largest root of f, and 0 is indeed a root of f and, since ηα is an increasing function,f is an increasing function on the domainη0,∞. It follows that the only root of the functionfabove is zero if and only iff0>0. Observe that

fx M−φx

M−a− _∞

0

ye^−xyν dy

,

2.23

where the interchange of the diﬀerentiation and integration in the second equation is permissible using the Lebesgue dominated convergence theorem, since for eachx≥ 0, y ≥ 0, ye^−xy < yand _∞

0 yνdy E₀I1−a < ∞. The rest of the proof follows sincef0 M−E0I1.

Proof ofb. To prove part b, first we observe that fx is an increasing function in its argument, and hencefxis a convex function in its argument. Sincefx η⁻¹x, it follows thatηαis a concave function.

Proof ofc. If the proof of partcfollows since, from2.20,W_τ^∗ <∞ almost everywhere if and only ifη0 0, in this case,ExW_τ^∗ x−τη0 x−τ/f0 x−τ/M− E₀I1.

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Remark 2.7. The equation given in partcofLemma 2.6is consistent with the well-known fact about the expected busy period of the M/G/1 queue.

LetU^∗^αbe the potential of the process X^∗. To findU^∗^α, we first need to introduce the following definition.

Definition 2.8. A L´evy process is said to be spectrally positivenegativeif it has no negative positivejumps.

Clearly, a L´evy processLis spectrally positive if and only if the process−L is spectrally negative. Furthermore, the processI^∗is spectrally positive with bounded variation.

Forθ, t∈R, we have

E

e^−θ

I∗t

e^tψθ, 2.24

where

ψθ Mθ−φθ. 2.25

We note that the functionηis the right-hand inverse of the functionψ.

We now define theα-scale function, which plays a major role in the applications of spectrally positivenegativeL´evy processes. This function is closely connected to the two- sided exit problem of such processescf. Bertion10.

Definition 2.9. Forα≥0, theα-scale functionof the processI^∗tW^α :R → Ris the unique function whose restriction toRis continuous and has Laplace transform

_∞

0

e^−θxW^αxdx 1

ψθ−α, θ > ηα, 2.26

and is defined to be identically zero on the interval−∞,0.

Lettingα0, we get the 0-scale function, which is referred to as the “scale function”

in the literature. We denote this function byW instead of W⁰throughout. We note that ψθ Mθ−φθ Mθ−aθ−_∞

0 1−e^−αxνdx Nθ−θ_∞

0 e^−αxνx,∞dx, whereN M−a >0. Letμ _∞

0 xνdx _∞

0 νx,∞dx. For everyx∈R, letFx _x

0 νy,∞dy/μ be the equilibrium distribution function corresponding toν. Letρ μ/Nand assume that ρ <1. It follows that

Wx 1

N ∞ k0

ρ^kF^kx, 2.27

whereF^kis thekth convolution ofF. Furthermore, we note that forα, x∈R, W^αx ^∞

k0

α^kW^k1x, 2.28

whereW^kis the kth convolution ofW.

We are now in a position to state and prove a lemma that characterizesU^∗^α.

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Lemma 2.10. U^∗^αis absolutely continuous with respect to the Lebesgue measure onτ,∞, and its density is given as follows:

U∗^α x, y

e^{−ηαx−τ}W^α y−τ

−W^α x−y

, x, y∈τ,∞. 2.29

Proof. Define the process^∧Ito be equal to− I^∗; it follows thatI^∧is a spectrally negative L´evy process. Fora, b∈R, we let

T_binf

t≥0 :I^∧t≥b

,

T_a⁻inf

t≥0 :I^∧t≤a

.

2.30

Supurn11proved thatforb >0theα-potential of the process obtained by killing the processI^∧atT_b∧T₀⁻ is absolutely continuous with respect to the Lebesgue measure on 0, b, and its density is equal to

W^αxW^α b−y

W^αb −W^α x−y

, x, y∈0, b. 2.31

It follows that, fora, b∈R,a < b, theα-potential of the process obtained by killing the process I∧ atT_b∧T_a⁻is absolutely continuous with respect to the Lebesgue measure ona, b, and its density is equal to

W^αx−aW^α b−y

W^αb−a −W^α x−y

, x, y∈a, b. 2.32

From Lemma 4 of Pistorious12, we haveW^αx Oe^ηαx asx → ∞. Lettinga → −∞

in the last density above, then the theαpotential of the process obtained by killing the process I∧ atT_b is absolutely continuous with respect to the Lebesgue measure on−∞, b, and its densitydenoted byu^α_bx, yis as follows:

u^α_b x, y

e⁻^ηα^b−xW^α b−y

−W^α x−y

, x, y∈−∞, b. 2.33

Observe that for anyA⊂τ,∞andx∈τ,∞,

P_x{X^∗_t ∈A}P _∗

I_t∈A, t < W_τ^∗ |I^∗₀x

P

I_t∈ −A, t < T_−τ |I₀ −x .

2.34

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Thus,

u∗^α x, y

u^α_−τ

−x,−y e⁻^ηαx−τW^α

y−τ

−W^α y−x

, x, y∈τ,∞.

2.35

It is seen that, forx≥τ,

C^α_g∗M, x, τ U^∗^αg^∗ x _∞

τ

g^∗ y ^∗

U^α x, dy

,

C_g⁰∗M, x, τ U^∗⁰g^∗ x _∞

τ

g^∗ y ^∗

U⁰ x, dy

.

2.36

Theorem 2.11. For anyα≥0 andx≥0, aforx≤λ,

Ex

exp

−αT^∗0

Mηαexp

−ηαx−τ

λ−x,∞exp

−zηα

U^αdz, 2.37

bforx > λ,

E_x

exp

−αT^∗0

exp

−ηαx−τ

. 2.38

Proof ofa. Letbe the sigma algebra generated byWλ, I_W_λ, then we have E_x

exp

−αT^∗0

E_x

exp

−α

W_λ _∗

T0−W_λ

Ex

exp

−α

Wλ _∗

T0−Wλ

|

E_x

exp−αWλEI_Wλexp−αW_τ^∗ Ex

exp−αWλexp

−ηαIWλ−τ E₀

exp−αW_λ−xexp

−ηαIWλ−xx−τ exp

−ηαx−τ E₀

exp−αWλ−xexp

−ηαIWλ−x

αφ

ηα

exp

−ηαx−τ

λ−x,∞exp

−zηα U^αdz

Mηαexp

−ηαx−τ

λ−x,∞exp

−zηα

U^αdz,

2.39

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where the third equation follows from the second equation, since given , T^∗0 − W_λ W_τ^∗ almost everywhere, the fourth equation follows from2.20above, the seventh equation follows from8Alili and Kyprianou9, and the last equation follows from2.21above.

Proof ofb. The proof of the partbof the theorem follows from2.20, since forx > λ,W_λ 0 andT^∗0W_τ^∗almost everywhere.

3. The Total Discounted, Long-Run Average Costs and the Stationary Distribution of the Dam Content

We now discuss the computations of the cost functionals using the total discounted cost as well as the long-run average cost criteria. LetWbe the length of the first cycle, that is,W T∗1−T^∗0, and letC_αx be the expected cost during the interval0,T^∗0, whenZ₀x. Since the content processZ is a delayed regenerative process with regeneration pointsT^∗0,T^∗1, . . ., using the delayed regeneration property, it follows that the total discounted cost associated with anP_λ,τ^M policy is given by

Cαλ, τ Cαx Ex

exp

−αT^∗0

EτCα1 1−E_τ

exp−αW , 3.1

whereC_α1is the total discounted cost during the interval0, W. From the definitions of Cαx, it follows that, forx > λ,

Cαx M

K1−REx

_W_τ^∗

0

e^−αtdt

C^α_g∗M, x, τ. 3.2

To compute C_αx for x ≤ λ, we let be the sigma algebra generated by Wλ, I_W_λ and proceed as follows:

C_αx M

⎧⎨

⎩K₂K₁E_x e^−αW^λ

−RE_x _T^∗₀

Wλ

e^−αtdt

⎫⎬

⎭ Ex

_W_λ

0

e^−αtgZtdtEx

_T^∗₀

Wλ

e^−αtg^∗Ztdt

M

K₂K₁E_x e^−αW^λ

−R α

E_x e^−αW^λ

−E_x

e^−α^T^∗⁰

E_x _W_λ

0

e^−αtgItdtE_x _T^∗₀

Wλ

e^−αtg^∗Ztdt

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M

K2K1Ex

e^−αW^λ

−R α

Ex

e^−αW^λ

−Ex

e^−α

T∗0

C_g^α0, τ, λ E_xE_x

⎛

⎝_T^∗₀

Wλ

e^−αtg^∗Ztdt|

⎞

⎠ M

−R α

E_x e^−αW^λ

−E_x

e^−α^T^∗⁰

C_g^α0, τ, λ Ex

⎡

⎣e^−αW^λEI_Wλ

⎛

⎝_W^∗_τ

0

e^−αtg^∗I_t^∗dt

⎞

⎠

⎤

⎦ M

−R α

E_x e^−αW^λ

−E_x

e^−α^T^∗⁰

C_g^α0, τ, λ E_x

e^−αW^λC^α_g∗M, IWλ, λ ,

3.3

where the second equation follows from the definition of the processZ, the third equation follows from the definition ofC_g^α0, τ, λ, the fourth equation follows from the definition of the content processZ and since, given,T^∗0 −W_λ W_τ^∗ almost everywhere, and the last equation follows from the definition of C^α_g∗M, x, λ.

We note that

E_τC_α1 C_ατ. 3.4

The following lemma shows howE_τexp−αW given in 3.1 can be computed and also gives a formula for computing the expected value ofW, which we will need later on to compute the long-run average cost.

Lemma 3.1. LetWbe the length of the first cycle as defined above, then a

E_τ e^−αW

1−Mηα _λ−τ

0

exp

−zηα

U^αdz, 3.5

b

EτW ME0Wλ−τ

M−E₀I1 ifE0I1< M, ∞ otherwise.

3.6

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Proof ofa. We note that, givenZ0τ,T₀^∗Walmost everywhere. Thus, for eachα≥0, E_τ

e^−αW E_τ

e^−αT⁰^∗ Mηα

λ−τ,∞exp

−zηα U^αdz

Mηα _∞

0

exp

−zηα

U^αdz− _λ−τ

0

exp

−zηα U^αdz

Mηα 1

Mηα− _λ−τ

0

exp

−zηα U^αdz

1−Mηα _λ−τ

0

exp

−zηα

U^αdz,

3.7

where the second equation follows2.37upon substitutingτforx, the third equation follows from the definition of theU^αand2.21.

Proof of (b). From3.5, it is evident that, starting atτ,W is finite almost everywhere if and only ifη0 0. From partaofLemma 2.6, it follows thatWis finite almost everywhere if and only ifE0I1< M. From2.14and3.5, we have

EτW Mη0E0W_λ−τif E0I1< M, ∞ otherwise.

3.8

The proof ofbis complete, since as shown in the proof of partcofLemma 2.6

η0 1

M−E₀I1 ifE0I1< M. 3.9 Now, we turn our attention to computing the long-run average cost per a unit of time.

LetM−E0I1 M^∗ and assume thatM^∗ > 0. From3.1,3.3, and3.4, it follows, by a Tauberian theorem, that the long-run average cost per unit of time, denoted byCλ, τ, is given by

Cλ, τ MK RE₀W_λ−τ C_g0, λ, τ E_τ

C_g^∗M, IWλ, τ

E_τW −RM

KM^∗ M^∗/M

Cg0, λ, τ Eτ

Cg^∗M, IWλ, τ

E₀Wλ−τ −RE0I1,

3.10

where KK₁K₂, and the second equation follows from3.6and the first equation.

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Remark 3.2. Assume that both penalty functions g and g^∗ are identically zero on their domains, andM^∗defined above is greater than zero. The following follows from3.10above:

Cλ, τ KM^∗

E0Wλ−τ−RE0I1. 3.11

LettingR 0, K 0 andgx I_τ,zx, x ∈0, λ andg^∗x I_τ,zx, x ∈ τ,∞ in 3.10, we get the following proposition which generalizes the results obtained by Lee and Ahn 3, where they assumed that the input process is a compound Poisson process and τ 0.

Proposition 3.3. Assume that M > E₀I1. Let Z lim_t_{→ ∞}Z_t, and,Hz be the distribution function of the processZ, then, forz∈τ,∞,

Hz M^∗

M E₀

W_λ∧z−τ E₀Wλ−τ

M^∗ M

Eτ

_∗ U

0 Iτ,zIWλ

E₀Wλ−τ . 3.12

4. Special Cases

In this section, we give the basic identities needed to compute the cost functionals when the input process is an inverse Gaussian process and a compound Poisson process, respectively.

Case 1. Assume that I is an inverse Gaussian process with transition function defined for x≥0, y≥0, μ >0, andσ²>0, by

p t, x, y

t σ

* 2π

y−x₃ exp

− μ

y−x

−t2

2 y−x

σ²

, y≥x.

0 y < x.

4.1

It follows that the processIis an increasing L´evy process with state spaceR, L´evy measure

ν dy

1 σ+

2πy3e^−yμ²^/2σ², 4.2

and L´evy component

φα

*

2ασ²μ²−μ

σ² . 4.3

Furthermore,E₀I1 1/μ.

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Substituting this L´evy component above in2.21, it is seen that the solution of this equation is as followswe omit the proof:

ηα α M

1−Mμ *

2αMσ²

1−Mμ₂

M²σ² . 4.4

To find theα-potential of the processI, for eachx ≥ 0 andβ ≥ 0, we definefβx exp−βx, and it is easily seen that

U^αf_β0 σ² ασ²*

2βσ²μ²−μ. 4.5

Throughout we letϕ_Z·be as the standard normal density function and let erf and erfc be the well-known error and complimentary error functions, respectively. Inverting the above function with respect toβ, we have

U^α dy

σ

√yϕ_Z √yμ

σ

dy

,μ−ασ² 2

-

e^αyασ²^/2−μerfc +

yασ²−μ

√2σ²

dy

u^α y

dy,

4.6

where

u^α y

σ

√yϕ_Z √yμ

σ

,μ−ασ² 2

-

e^αyασ²^/2−μerfc +

yασ²−μ

√2σ²

. 4.7

From2.13, it follows that, forx≤λ, E_x

exp−αWλ

αU^αI_λ−x,∞0

ασ²−μ

ασ²−2μe^αλ−xασ²^/2−μerfc,+

λ−xασ²−μ

√2σ² -

− μ

ασ²−2μerfc

,√λ−xμ

√2σ² -

,

4.8

where the last equation follows by integratingU^αdy over the intervalλ−x,∞.

Inverting the right hand side of4.8with respect toα, it follows that, givenI0x≤λ, the distribution function ofWλdenoted byFWλ is given by

F_W_λt 1 2erfc

λ−xμ−t

√2σ²

− 1

2e^2μt/σ²erfc

λ−xμt

√2σ²

, t≥0. 4.9

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Furthermore, forx≤λ, ExWλ U⁰I_0,λx

σ _λ−x

0

√1yϕZ+ yμ

σ

dyμ 2

_λ−x

0

erfc ,

− .y

2 μ σ

- dy

λ−xμ

2 σ+

λ−x ϕ_Z+ λ−xμ

σ

λ−xμ²σ²

2μ erf

⎛

⎝ /

λ−x 2

μ σ

⎞

⎠,

4.10

where the third equation follows from the second equation upon tedious calculations which we omit.

We now turn our attention to computing the distribution function ofI_W_λ denoted byF_I_Wλx. We first need the following identity which expresses the L´evy componentφα given in4.3in a form suitable for computingFI_Wλ. The proof of this identity follows from 4.3after some simple algebraic manipulations which we omit:

φα

*

2ασ²μ²−μ σ² 2

σ² α

φα−μ

.

4.11

For eachβ∈R, we write _∞

λ

e^−βxFI_Wλxdx φ β β

_∞

λ

e^−αxu⁰xdx

2 σ²

1 φ

β _∞

λ

e^−βxu⁰xdx− μ β

_∞

λ

e^−βxu⁰xdx

2 σ²

_∞

0

e^−βxu⁰xdx _∞

λ

e^−βxu⁰xdx−μ β

_∞

λ

e^−βxu⁰xdx

2 σ²

_∞

λ

e^−βx x

λ

u⁰

x−y

−μ u⁰

y dy

dx

,

4.12

where the first equation follows from the second equation given the proof of part a of Lemma 2.4 by letting x 0, the second equation follows from 4.11, the third equation follows from4.5upon lettingα0, and the fourth equation follows from the third equation through integration by parts.

From4.12, it follows that, for eachx≥λ,

FI_Wλx 2 σ²

_x

λ

u⁰

x−y

−μ u⁰

y dy

. 4.13

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Case 2. Assume thatI is an increasing compound Poison process with intensityuandF as the distribution function of the size of each jump. This model is treated in details in references 3,4,6. Here, we give the basic entities involved when the drift termsa 0. For the proof of these entities and more in depth analysis of this case, the reader is referred to the above- mentioned references.

It is obvious that

φα u _∞

0

1−e^−αx

Fdx, 4.14

E0I1 uμ, whereμ is the expected jump size of the compound Poisson process.

Define, for anyα≥0 andy≥0,Fαy u/uαFy. Forn∈N, we letF_αⁿy be thenth convolution of F_αy, where F_α⁰y 1 for ally ≥ 0. For eachy ≥ 0, we define R_αy 0_∞

n0F_αⁿyto be the renewal function corresponding toF_αy. It follows that

U^α dy

1 uαRα

dy

. 4.15

Furthermore, forx≤λ,

E_x

exp−αWλ

1−αU^αI_0,λ−x0 1− α

uαR_αλ−x, ExWλ U⁰I_0,λ−x0

1

uR₀λ−x.

4.16

Also,

E0

e^−αI^Wλ

φα _∞

λ

e^−αxU⁰dx

_∞

λ

e^−αxR0dx− _∞

0

e^−αxFdx _∞

λ

e^−αxR0dx

_∞

λ

e^−αxR0dx− _∞

λ

e^−αx _x

λ

F dx−y

R0

dy ,

4.17

where the first equation follows from the second equation in the proof of part a of Lemma 2.4, by lettingx0. Furthermore, the second equation follows from4.14and4.15.

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Inverting4.17with respect toα, the distribution function ofIWλ, denoted byG, is given through

Gdx

R₀dx− _x

λ

F dx−y

R₀

dy

I_λ,∞x

Fdx _x

λ

F dx−y

R₀ dy

I_λ,∞x.

4.18

Acknowledgment

This Research was supported, in part, by a 2010 Summer Research Grant from the College of Business and Economics, UAE University.

References

1 Y. Lam and J. H. Lou, “Optimal control of a finite dam: Wiener process input,” Journal of Applied Probability, vol. 24, no. 1, pp. 186–199, 1987.

2 M. Abdel-Hameed and Y. Nakhi, “Optimal control of a finite dam usingP_λ,τ^Mpolicies and penalty cost:

total discounted and long run average cases,” Journal of Applied Probability, vol. 28, no. 4, pp. 888–898, 1990.

3 E. Y. Lee and S. K. Ahn, “P_λ^M-policy for a dam with input formed by a compound Poisson process,”

Journal of Applied Probability, vol. 35, no. 2, pp. 482–488, 1998.

4 M. Abdel-Hameed, “Optimal control of a dam usingP_λ,τ^M policies and penalty cost when the input process is a compound Poisson process with positive drift,” Journal of Applied Probability, vol. 37, no.

2, pp. 408–416, 2000.

5 J. Bae, S. Kim, and E. Y. Lee, “AP_λ^M policy for M/G/1 queueing system,” Applied Mathematical Modelling, vol. 26, pp. 929–939, 2002.

6 J. Bae, S. Kim, and E. Y. Lee, “Average cost under theP_λ,τ^M policy in a finite dam with compound Poisson inputs,” Journal of Applied Probability, vol. 40, no. 2, pp. 519–526, 2003.

7 R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York, NY, USA, 1968.

8 J. Lamperti, Stochastic Processes: A survey of the Mathematical Theory, Springer, New York, NY, USA, 1977.

9 L. Alili and A. E. Kyprianou, “Some remarks on first passage of L´evy processes, the American put and pasting principles,” The Annals of Applied Probability, vol. 15, no. 3, pp. 2062–2080, 2005.

10 J. Bertion, L´evy Processes, Cambridge University Press, Cambridge, UK, 1996.

11 V. N. Supurn, “The ruin problem and the resolvent of a killed independent increment process,”

Ukrainian Mathematical Journal, vol. 28, pp. 39–45, 1976.

12 M. R. Pistorius, “On exit and ergodicity of the spectrally one-sided L´evy process reflected at its infimum,” Journal of Theoretical Probability, vol. 17, no. 1, pp. 183–220, 2004.

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Policies When the Input Process Is a Nonnegative L ´evy Process

Research Article

Control of Dams Using P

Policies When the Input Process Is a Nonnegative L ´evy Process

Mohamed Abdel-Hameed

1. Introduction and Summary

2. Basic Results

3. The Total Discounted, Long-Run Average Costs and the Stationary Distribution of the Dam Content

4. Special Cases

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Algebra

Decision Sciences

Discrete Mathematics

Stochastic Analysis