On a storage process for fluid networks with multiple L evy input - PDF document

On a storage process for fluid networks with multiple L´ evy input Krzysztof D¸ ebicki University of Wroc� law, Poland Presentation at EVA 2005 Gothenburg, Sweden August, 2005 1

Outline of the talk • Two-node tandem network (Mandjes, van Uitert, K.D.) – New representation for Q 2 – L´ evy case: distribution of Q 2 – Examples • n-node tandem network (Dieker, Rolski, K.D.) – Skorokhod problem – Stationary representation – Laplace transform 2

Two-node tandem network • r 1 > r 2 > 0 • J ( t ) - with stationary increments, E J (1) < r 2 We are interested in P ( Q 2 > u ) • Q 2 - stationary buffer content at the second node - Kella, Whitt, Rubin, Shalmon, Mandjes, van Uitert,... 3

Representation for Q 2 Following Reich’s representation we have Q 1 = d sup { J ( t ) − r 1 t } t ≥ 0 and Q total = d sup { J ( t ) − r 2 t } . t ≥ 0 Hence Q 2 = d sup { J ( t ) − r 2 t } − sup { J ( t ) − r 1 t } . t ≥ 0 t ≥ 0 4

New representation for Q 2 Let u t u = . r 1 − r 2 Theorem 1. For each u ≥ 0 , � � P ( Q 2 > u ) = P sup { J ( t ) − r 2 t } − sup { J ( t ) − r 1 t } > u . t ∈ [ t u , ∞ ) t ∈ [0 ,t u ] This representation enables us to analyze the distribution of Q 2 for following classes of input processes: • processes with independent increments • Gaussian processes 5

Input with stationary independent increments Theorem 2. Let { J ( t ) , t ∈ R } be a stochastic process with stationary independent increments and let µ = E J (1) < r 2 . Then for each u ≥ 0 , and J 1 ( · ) and J 2 ( · ) independent copies of the process J ( · ) , � � P ( Q 2 > u ) = P sup { J 1 ( t ) − r 2 t } > sup {− J 2 ( t ) + r 1 t } . t ∈ [0 , ∞ ) t ∈ [0 ,t u ] 6

Input with spectrally positive L´ evy process Let J ( t ) be a spectrally positive L´ evy process. Introduce θ ( s ) := log( E e − s ( J (1) − r 1 ) ) . Theorem 3. Let { J ( t ) , t ∈ R } be a spectrally positive L´ evy process with µ := E J (1) < r 2 . Then, for each x > 0 , θ − 1 ( x ( r 1 − r 2 )) E e − xQ 2 = r 2 − µ · x − θ − 1 ( x ( r 1 − r 2 )) . r 1 − r 2 Remark 1. Theorem 3 can be considered as an analogue of the result of Zolotarev who obtained the Laplace transform of P ( Q 1 < u ) for J ( · ) being a spectrally positive L´ evy process. 7

Pollaczek-Khintchine representation Theorem 4. Let { J ( t ) , t ∈ R } be a spectrally positive L´ evy process with µ := E J (1) < r 2 . Then ∞ � ̺ i − 1 H ⋆i ( u ) , P ( Q 2 ≤ u ) = (1 − ̺ ) i =1 where • ̺ := ( r 1 − r 2 ) / ( r 1 − µ ) • H ( · ) is a distribution function such that H ( x ) = 0 for x < 0 and � ∞ e − xv d H ( v ) = θ − 1 ( x ) ̺x 0 for x ≥ 0 . 8

Examples: exact distributions • J ( t ) is a standard Brownian motion. � r 1 − 2 r 2 √ u � �� P ( Q 2 > u ) = r 1 − 2 r 2 e − 2 r 2 u √ r 1 − r 2 1 − Ψ r 1 − r 2 √ u r 1 � r 1 � + Ψ √ r 1 − r 2 , r 1 − r 2 where Ψ( x ) = P ( N > x ) . � If c 1 > 2 c 2 , then P ( Q 2 > u ) ∼ r 1 − 2 r 2 e − 2 r 2 u . r 1 − r 2 � If c 1 ≤ 2 c 2 , then r 2 � � 1 1 1 √ u exp P ( Q 2 > u ) ∼ − 2( r 1 − r 2 ) u . � 2 π ( r 1 − r 2 ) Also we can get the exact distribution if • J ( t ) is a Poisson process. 9

Examples: asymptotic results • J ( t ) = X α, 1 ,β ( t ) with α ∈ (1 , 2) and β ∈ ( − 1 , 1] . Then C 1 u 1 − α ≤ P ( Q 2 > u ) ≤ C 2 u 1 − α as u → ∞ . • J ( t ) = X α, 1 , 1 ( · ) with α ∈ (1 , 2) . Then P ( Q 2 > u ) ∼ � 1 − α � 1 1 r 1 u 1 − α . ∼ Γ(2 − α ) cos( π ( α − 2) / 2) r 2 r 1 − r 2 Also we can get the asymptotic for • J ( t ) compound Poisson input, with regularly varying jumps 10

n-node tandem network • J ( t ) = ( J 1 ( t ) , ..., J n ( t )) ′ - n -dimensional L´ evy process with mutually independent components and J 1 ( t ) is a spectrally positive L´ evy process, J 2 ( t ) , . . . , J n ( t ) are subordinators • r = ( r 1 , ..., r n ) ′ - output rates • P = ( p ij ) n i,j =1 - routing matrix; 0 < p ii +1 ≤ 1 and p ij = 0 if j � = i + 1 Moreover, we tacitly assume that p ii +1 > r i +1 N1 (Work-conserving) r i , ( I − P ′ ) − 1 E J (1) < r . N2 (Stability) 11

n-node tandem network, ctd. We are interested in the transient joint distribution of • Q ( t ) = ( Q 1 ( t ) , ..., Q n ( t )) ′ - storage process • B ( t ) = ( B 1 ( t ) , ..., B n ( t )) ′ - running busy period process, where B i ( t ) = t − sup { 0 ≤ s ≤ t : Q i ( s ) = 0 } . Q ( t ) is the unique solution of the Skorokhod problem of J ( t ) − ( I − P ′ ) rt with reflection matrix I − P ′ , that is S1 Q ( t ) = w + J ( t ) − ( I − P ′ ) rt + ( I − P ′ ) L ( t ) , t ≥ 0 , S2 Q ( t ) ≥ 0 , t ≥ 0 and Q (0) = w , S3 L (0) = 0 and L is nondecreasing, and � ∞ S4 � n 0 Q i ( t ) dL i ( t ) = 0 . i =1 12

n-node tandem network, ctd. Proposition 1. Suppose that Q ( t ) is the storage process associated to the stochastic network ( J, r, P ) . Then ( I − P ′ ) − 1 Q ( t ) is the solution to the Skorokhod problem of X ( t ) = ( I − P ′ ) − 1 J ( t ) − rt with reflection matrix I . Theorem 5. The stationary distribution ( W ( ∞ ) , B ( ∞ )) is the same as the distribution of (( I − P ′ ) X, G ) , where X = ( X 1 , ..., X n ) ′ and G = ( G 1 , ..., G n ) ′ with     i k − 1 � �  J k ( t ) − r i t X i = sup p jj +1    t ≥ 0 k =1 j =1 G k = sup { t ≥ 0 : X k ( t ) = X k ( t ) } . 13

n-node tandem network, ctd. Theorem 6. Consider a tandem stochastic network ( J, r, P ) that N1-N2 hold. Then for α = ( α 1 , ..., α n ) ′ > 0 , β = ( β 1 , ..., β n ) ′ > 0 E e − <α,Q ( ∞ ) > − <β,B ( ∞ ) > = = E e − α n X n − β n G n × n − 1 E e − α j X j − [ � n ℓ = j +1 Ψ J ℓ ( α ℓ )+ � n ℓ = j +1 ( p ℓ − 1 ℓ r ℓ − 1 − r ℓ ) α ℓ + � n p = j β p ] G j � × p = j +1 β p ] G j , E e − p jj +1 α j +1 X j − [ � n ℓ ( α ℓ )+ � n ℓ = j +1 ( p ℓ − 1 ℓ r ℓ − 1 − r ℓ ) α ℓ + � n ℓ = j +1 Ψ J j =1 where X ( t ) = ( I − P ′ ) − 1 J ( t ) − rt � E e − λJ i (1) � Ψ J i ( λ ) = − log . The formula can be made explicit by the use of fluctuation identities. But is a bit long... 14