Study on Markov Alternative Renewal Reward. Process for VLSI Cell Partitioning

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 40, HIKARI Ltd, Study on Markov Alternative Renewal Reward Process for VLSI Cell Partitioning R. Manikandan *, P. Swainathan * and V. Vaithiyanathan * * School of Coputing, SASTRA University, Thanavur , India anikandan75@core.sastra.edu, deanpsw@sastra.edu, vvn@it.sastra.edu Copyright 2013 R. Manikandan et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. Abstract. In this paper, we introduce a constructive approach to define Markov Alternative Renewal Reward processes(marp).the construction is based on Poisson process, which is siple and intuitive. The construction akes it easy to interpret the paraeters of Markov Renewal reward processes. Also it akes to establish basic equations such as Kologorov differential equations, atheatically rigorously for Markov Alternative Reward process(marp). In addition, the approach can be used to construct alternative Markov chains with a finite nuber of states. Keywords: Markov chain, Markov alternative process, Matrix-analytic ethods 1. Introduction Markov Alternative Renewal Reward process (MARP) is a popular tool for odeling arrival processes of stochastic systes such as queuing systes, reliability systes, telecounication networks, VLSI Design, inventory and supply chain systes and risk and insurance systes. Alternative renewal reward processes(marp) coes fro (i) Its versatility in odeling stochastic processes; (ii) Its Markovian property that leads to Markovian structures; and (iii)the Maneuverability in the resulting Markov chains.

2 1950 R. Manikandan, P. Swainathan and V. Vaithiyanathan Counting processes are iportant stochastic processes for Science and Engineering. In order to capture the characteristics of real stochastic processes, a nuber counting processes have been introduced. Soe well known Counting processes are Poisson process, Copound Poisson process, Markov Modulated Poisson process [2]. An interesting and useful idea to generate counting processes is by odeling the transpositions of Markov chains [11]. By utilizing that idea in a systeatic anner, Neuts introduced Markov arrival processes as generalizations of Poisson processes, Copound Poisson processes and Markov odulated Poisson processes [6],[7],[8]. The focus of this paper is on a constructive way to define alternative approach of Markov Renewal Reward Processes. The new approach is based on Poisson processes only. With the new definitions, it is easy to understand paraeters of Markov Alternative Renewal Reward Processes(MARP) intuitively. Although it is siple and intuitive, the new definition is atheatically rigorous. Basic results on Markov alternative processes, such as the well known (generalized) Kologorov differential equations can be proved rigorously based on the new definition and soe eleentary properties on exponential distributions. Therefore, the new definition akes it easy to introduce Markov Alternative Renewal Reward Processes(MARP) and to use Markov arrival processes in Stochastic odeling for both researchers and practitioners. The rest of the paper is organized as follows. In section 2, exponential distributions, Poisson processes and their properties are introduced. A constructive way to define alternative Markov Chains ( MARC) with a finite nuber of phases is introduced in section 3.Section 4 gives the definitions of Markov Alternative Renewal Reward Processes(MARP) and Batch Markov Alternative Renewal Reward processes(marp).also distribution of tie between arrivals are established with suitable exaple. 2. Preliinaries In this section, We define exponential distributions and Poisson processes. Properties used in later sections are collected. All the properties can be proved by routine calculations. We refer to [10] for proofs of the properties. 2.1 Definition: A Non negative rando variable x has an exponential distribution if its probability distribution function is given by Ft ( ) = px { t} = 1 exp{ λt}, t (2.1) Where λ is a positive real nuber. We call X an exponential distribution with paraeter λ. Properties of exponential distributions used in this paper are collected. (i) Assue that X has on exponential distribution with paraeter λ then

3 Study on Markov alternative renewal reward process 1951 p{ X > t+ s/ X > s} = p{ X > t} holds for t 0 and s 0, which is called the eory less property. The eory less property says that the distribution of the residual tie X s, given X > s, denoted by X s/ X > s, is independent of the tie s that has elapsed. (ii) Assue that X 1, X 2 and X 3 are three independent exponential rando Variables with paraeters λ1, λ2 and λ 3. We have for sall t, 2 2 p{ X1+ X2 t} = 0.5 λλ 1 2t + Ot ( ) = Ot ( ); p{ X1 = in{ X1, X2} t, X 1+ X3 > t} = λ1t+ O( t) (2.2) (iii)assue that { X,1 n) are independent exponential rando variables With paraeters { λ,1 n}, respectively. Then X = in{ X1.... X n } is exponentially distributed with paraeter λ1+ λ λn (iv) Assue that { X,1 n} are independent exponential rando Variables with paraeters { λ,1 n}, respectively. Then px { 1 = in{ X1... X n}} = λ1/ ( λ1+... λn). (v) Assue that { Nt ( ), t 0} is a counting process. Fig 2.1 : X=in{X 1,X 2 } and its residual at tie S for n=2 Definition 2.2 : A counting process { Nt ( ), t 0} is called a Poisson process if { Nt ( ) n} = { X1+ X X n + Xn+ 1 > t}, forn 0 and t 0, where {X 1, X X n,...} are independent exponential rando variables with paraeter λ.

4 1952 R. Manikandan, P. Swainathan and V. Vaithiyanathan Fig 2.2 A saple path of a Poisson process For a Poisson process { Nt ( ), t 0} with paraeter λ, We have (i) N(0)=0 (ii) E{ Nt ( )} = λt Paraeter λ in the average nuber of events per unit tie. (iii) Let Y be the tie elapsed until the first event after tie t (Fig 2.3).Then Y has an exponential distribution with paraeter λ. This is called eory less property of Poisson processes. Fig 2.3 Meory less property of Poisson process. 3.1 Construction of Markov Alternative Renewal Reward process (MARP) In this section, We construct alternative Markov Renewal Reward Processes on Poisson processes defined in section 2. Definition 3.1.1: Let { αi, 1 i } be non negative nubers with a unit su { i,, α α = 1), {a i,1 i } non negative real nubers and a finite positive integer ( 2).

5 Study on Markov alternative renewal reward process 1953 Assue qi,1 i, i 1 qi, i+ 1 i, +... q q > 0, for 1 i. A stochastic process {(), It t> 0} on phases {1,2,... } is defined as follows 1. Define (-1) independent Poisson processes with paraeters { q,1 i } of q = 0, the corresponding Poisson process has no i, i event at all. 2. Deterine I(0) by probability distribution { αi,1 i }. 3. A tie t o, if I(t)=i, then I(t) stays in phase I until the first event occurs in the -1 Poisson processes corresponding to { q,1, i}, for 1 i. i, Fig 3.1 Poisson processes and corresponding Markov alternative Reward Processes Proposition 3.1.1: For the Markov alternative Renewal reward {(), It t 0}, We have p 1 ( t) = p( t) Q=Qp(t), for t>0, and p(0)=i. Where I is the identity atrix. Proof: Let { X i,,1 i } be independent exponential rando variables with paraeter q i respectively. Denote by Jtt (, + δt) the nuber of transitions occurred in the interval (, tt+ δt). Conditioning on I(t), we have the following calculations.

6 1954 R. Manikandan, P. Swainathan and V. Vaithiyanathan pi ( t+ δ t) = p{() It = k/ I(0) = I} pit {( + δt) = Jtt, (, + δ) = n/ It () = ki, (0) = i} k= 1 n= (3.2) = p () t p{( I δt) =, J(0, δt) = n/ I(0) = k} i, k= 1 n= 0 = p ( t) p{ I( δt) =, J(0, δt) = 0 / I(0) = k} + p ( t) p{ I( δt) =, J(0, δt) = 1/ I(0) = K} ik, ik, k= 1 k= 1 + pik, () t p{( I δt) =, J(0, δt) = n/ I(0) = k} k= 1 n= 2 Using property (ii),(iii) and (iv), we obtain k = 1 p ( t) p{ I( δt) =, J(0, δt) = 0 / I(0) = k} ik, = pi, () t p{( I δt) =, J(0, δt) = 0/ I(0) = } = p() t 1 q t+ O( t) k, δ δ (3.3) k= 1, k= k = 1 p ( t) p{ I( δt) =, J(0, δt) = 1/ I(0) = k} ik, = p ( t) p{ X = in{ X, l} < δt, in{ X + X } > δt} (3.4) ik, k, k k, i, k= 1, k 1 l, l k 1 l, l = p ()( t q δt+ O( δt)) and k= 1, k ik, k, p ( t) φ{ I ( δ t) =, J (0, δ t) = n / I (0) = k} i, k k = 1 n= 2 k = 1 { δ 1 l, i k, l= } = p () t p in { X + X } t i, k k, l l, pik, () t p{ Xkl, Xl, δt} (3.5) + k= 1 1 l, l k, = O( δt) Cobining equations (3.2),(3.3),(3.4) and (3.5) gives

7 Study on Markov alternative renewal reward process 1955 p ( t+ δt) = p ( t)(1 + q δt) + p ( t) q δt+ O( δt) (3.6) i, i, i, i, k k, k= 1, k Which leads to the Kologorov forward differential equation dp, () i t = pik, () tqk,. The Kologorov backward differential equation can be dt k = 1 shown siilarly. Exaple 3.1.1: Let Nt ( ), t 0 be a Poisson (counting) process of rate λ. N(t) takes values in {0,1,2,... }.Further, we have for t 0, s 0 and fixing the values of Ns ( ) = i( ) p( Nt ( + s)) = /( Nu ( ), u< sns, ( ) = i) p( Nt ( + s) Ns ( ) = i/ Nu ( ), u sns, ( ) = i) p( Nt ( + s) Ns ( ) = i) p( Nt ( + s)) = / Ns ( ) = i Where the second quality follows by the independent increent property of the Poisson process. 3.2 Inspection paradox Suppose that the distribution of the tie between renewals F is unknown. one way to estiate it is to choose soe sapling ties t 1,t 2 etc and for each t i, record the total aount of tie between the renewals ust before and after t i. This schee will over estiate the inter renewal ties. For each sapling tie t, we will record X N(t)+1 =S N(t)+1 - S N(t). Find its distribution by conducting(conditioning) on the tie of the last renewal prior to tie t. p{ X > x} = E{ p{ X > x/ S = t s} N() t + 1 N()1 t + N() t if S>x then px { > N()1 t x/ S = N() t t + s} = 1 if S x then px { > x/ S = t s} = px { > x/ X > s} px { > x} 1 F( x) px { > s} 1 Fs ( ) N() t + 1 = = > N()1 t + N() t + 1 N() t N()1 t + N()1 t + 1 F( x)

8 1956 R. Manikandan, P. Swainathan and V. Vaithiyanathan px { > x} 1 F( x) px { > s} 1 Fs ( ) N() t + 1 = = > N()1 t + 1 F( x) For any S, px { > x/ S = t s} 1 Fx ( ) N()1 t + N() t so p{ X > x} = E{ P{ X > x / S = t s} 1 F( x) = p( X > x) N()1 t + N()1 t + N() t when X is an ordinary inter-renewal tie 4. Construction of Markov alternative Renewal process (MARP) AND Batch Markov alternative Renewal process (BMARP) In this section, We define soe special events, called alternatives, associated with phases and we keep track of the nuber of arrivals. Definition 4.1: Let { αi,1 i } be non negative nubers with a unit su, { d,1 i } and {d,1 i, } are non negative nubers and is a 0( i, ) 1( i, ) finite positive integer. d = d d + d +... d +... d > 0 Assue 0(, i ) 0(,1) i 0(, i i 1) 0(, i i+ 1) 0(, i ) 1(, i ) Fig 4.1 Saple paths of underlying Poisson processes I(t) and N(t) of Markov alternative process.

9 Study on Markov alternative renewal reward process 1957 Lea 4.1: For the process { Nt ( ), It ( ), t 0}, we have (i) The So orn tie of { Nt ( ), It ( ), t 0} in state (n,i) has an exponential distribution with paraeter d ( i, i) for 1 i 0 (ii) The probability that the next phase is and no arrival at the transition epoch is given by p0,( i, ) d0,( i, ) / d0,( i, ) given that the current stats is (n,i) for 1 i and n 0 (iii) The probability that the next phase is and an arrival occurs at the transition epoch is given by p 1,(, i ) d 1,(, i ) /( d 0,(, i ) ) given that the current state in (n,i) for 1 i, and n 0. Let D0 = ( d0,( i, ) ), D 1 = ( d1,( i, ) ) and D=D0 + D1, three axiu atrices. Proposition 4.1: For an Markov alternative Renewal processes { Nt ( ), It ( ), t 0} with a atrix representation (D 0,D 1 ), define n 1 p (,) z t = zp(,),for n t z 0 then 1 ( D0+ DZ 1 ) t p (,) z t = e n= 0 Proof: Based on proposition 4.1, the average nubers of arrivals per unit tie, called the arrival rate, can be found as λ = θde,, where θ is the stationary distribution of D (assuing that D is irreducible),(i.e) θ D = 0 and θe= Distribution of tie between arrivals The inter event distribution on the Markov alternative Renewal Processes(MARP) are of phase type. The tie fro an arbitrary tie to the first event has the ph representation ( θ, D0 ), the distribution of arbitrary interval has the representation ( φ, D0 ). Exaple 4.1: The MARP with the paraeter atrices D 0 λ 0 0 = 0 - λ λ 0 0 -μ 0 λ 0 D1 = μ 0 0

10 1958 R. Manikandan, P. Swainathan and V. Vaithiyanathan is an alternating renewal process. The alternating intervals are exponentially distributed respectively generalized Erlang-2 distributed. We find 1/ λ 0 0 = 0 0 1/ μ 1 ( D0 ) 0 1/ λ 1/ μ and ( ) D0 D1 = ~ 2 1 ~ θ = + λ λ μ φ = λ μ ( 1/,1/,1/ ) ( 1/2,1/2,0) the Laplace transfor of the distribution of an arbitrary inter arrival tie is ~ 1 λ 1 λ μ λ/2 μ λ/2( s + 2 μ) H() s = + = 1+ = 2 s+ λ 2 s+ λ s+ μ s+ λ s+ μ ( s+ λ)( s+ μ) μ if we λ = 2μ we find this distribution to be. The Laplace transfor of an s + μ exponential distribution. Thus the tie between two arbitrary arrivals is exponential. Nevertheless, the process is not a Poisson process. The MARP of exaple 4.1 can be forulated as Markov Renewal with the kernel λt 0 1-e At ( ) = λ =H(t)p μt μ (4.2.1) λt 1 e e 0 λ+ μ μ λ with λt 1-e Ht ( ) = λ P= 0 1 μt μ λt e e (4.2.2) λ+ μ μ λ

11 Study on Markov alternative renewal reward process 1959 Conclusion We could forulate a Markov renewal process with the epheeral states too, but this will not be relevant in general. On could question why MARP foralis is introduced since the Markov renewal forulation is already well established and is slightly ore general than the MARP. The advantage of the MARP foralis is that it preserves of the Markov property of the process not only at arrivals but ore generally in tie. The inter arrival- tie of MARP can be extended to iniu delay VLSI Circuit partitioning probles. The forulae state of in this note in MARP forulis are generally less transparent and less explicit it expressed in ters of the Markov renewal process. References [1] Chakravarthy, S. R., The batch Markovian arrival process: A review and future work. In advances in probability and stochastic process, Notable publications, Inc New Jersey, 2001, [2] Cinlar, E., Markov Renewal theory, Adv. Appl.Prob. 1(1969), [3] He, Qi-Ming and M. E Neuts, Markov chains with arked transitions, Stochastic processes and their applications, 74(1), 1998, [4] Latouche, G. and V. Raaswai, Introduction to atrix analysis ethods in stochastic odeling, ASA and SIAM, Philadelphia,USA,1999 [5] Lucantoni, D. M., New results on the single server queue with a batch Markovian arrival process, Stochastic odels,7(1991), [6] Neuts, M. F., A versatile Markovian point process, Journal of Applied probability, 16(1979), [7] Neuts, M.F., Matrix Geoetric solutions in stochastic odels- An algorith approach, The Johns Hopkins university press, Baltiore,1981. [8] Neuts, M.F., Models based on the Markovian arrival process, IEICE Trans. Coun. E75-B, 1992, [9] Raaswai, V., The N/G/1 queue and its detailed analysis, Adv.Appl.Prob., 12(1980),

12 1960 R. Manikandan, P. Swainathan and V. Vaithiyanathan [10] Ross, S. M., Introduction to Probability Models, Ninth Edition, Acadeic Press,2007. [11] Rudoo, M., Point processes generated by transitions of Markov chains, Adv. Appl. Prob. 5(1973), Received: June 5, 2013