Queueing for ergodic arrivals and services

Queueing for ergo dic arriv als and services L´ aszl´ o Gy¨ orfi ∗ Guszt´ av Morv ai † No v em b er 7, 2018 Abstract In this pap er w e revisit the results of Lo ynes (196 2) on stabilit y of queues for ergo dic arriv als and services, and show examples when the arr iv als are b oun ded and ergo dic, the service rate is constan t, and u n der stabilit y the limit distribution h as larger than exp onen tial t ail. ∗ Dept. o f Computer Science and Information Theory , T ec hnical University of Budap est, 1 521 Sto czek u. 2, Budap est, Hungary . E-mail:gyorfi@szit.bme.hu † Research Group for Informatics and E lectronics of the Hungaria n Academy of Sciences, 1 521 Sto czek u. 2, Budap est, Hungary . E-mail: mor v ai@szit.bme.hu 0 1 In tro ductio n The analysis of a que ueing model consists of t wo steps: stabilit y study and the c haracteriza- tion o f the limit distribution. In this pap er w e consider long range dep endence, i.e. ass ume ergo dic arriv a ls and services. Concerning stability revisit the res ult of Loynes (1962), who extended t he Mark o vian appro a c h in an elegant w a y . F or the weak dep enden t situation the limit distribution usually has an almost exp onen tial tail. Here w e show coun terexample for ergo dic situation, i.e. when the a rriv als a re b ounded and ergo dic, the service rate is constant. 2 Stabilit y 2.1 General result Let X 0 b e arbitrary random v ariable. Define X n +1 = ( X n + Z n +1 ) + (1) for n ≥ 0, where { Z i } is a sequence of rando m v ariables. W e are ineterested in the stabilit y of { X i } , i.e. we are lo oking for conditions on { Z i } under whic h X i has a limit distribution. F or the classical Mark o vian approach { Z i } are indep enden t a nd iden tically distributed, when stabilit y o f { X i } means that there exis ts a uniqu e limit distribution of X n . Here conside r the case when { Z i } is o nly stationary and ergo dic. F ollowing Lindv all (1992) in tro duce a stronger concept o f stabilit y: Definition 1 We s ay that the se quenc e { X i } is c o uple d with the se quenc e { X ′ i } if { X ′ i } is stationary and er go d ic a nd ther e is an almost s ur ely finite r andom variable τ such that X ′ n = X n for n > τ . Put V 0 = 0 , V n = n − 1 X i =0 Z − i , ( n ≥ 1) . Theorem 1 If { Z i } is stationary and er go dic, and E { Z i } < 0 , then { X i } is c o uple d w ith a stationary and er go d ic { X ′ i } such that X ′ 0 = sup n ≥ 0 V n . Pro of. Step 1. Le t X − N , − N = 0 and define X − N ,n for n > − N by the following recursion, X − N ,n +1 = ( X − N ,n + Z n +1 ) + for n ≥ − N . 1 W e show that X − N , 0 is monoton increasing in N , and almost surely , lim N →∞ X − N , 0 = X ′ , where X ′ = sup n ≥ 0 V n , and X ′ is finite a.s. Notice that X − N ,n +1 = ( X − N ,n + Z n +1 ) + for n ≥ − N . First we pro o v e t ha t for n > − N , X − N , − n = max { 0 , Z n , Z n + Z n − 1 , . . . , Z n + . . . + Z − N +1 } . (2) F or n = − N + 1, X − N , − N +1 = ( X − N , − N + Z − N +1 ) + = ( Z − N +1 ) + = max { 0 , Z − N +1 } . F or n = − N + 2, X − N , − N +2 = ( X − N , − N +1 + Z − N +2 ) + = max { 0 , X − N , − N +1 + Z − N +2 } = max { 0 , max(0 , Z − N +1 ) + Z − N +2 } = max { 0 , Z − N +2 , Z − N +2 + Z − N +1 } . No w w e pro ov e b y induction from n to n + 1. X − N ,n +1 = ( X − N ,n + Z n +1 ) + = max { 0 , max { 0 , Z n , Z n + Z n − 1 , . . . , Z n + . . . + Z − N +1 } + Z n +1 } = max { 0 , Z n +1 , Z n +1 + Z n , . . . , Z n +1 + . . . + Z − N +1 } . W e hav e completed the pro of of (2). Thu s X − N , 0 = max { 0 , Z 0 , Z 0 + Z − 1 , . . . , Z 0 + . . . + Z − N +1 } , whic h imlies that X − N , 0 is monoton increasing, since the maximu m is tak en ov er lar g er and larger set. It remains to pro v e that X − N , 0 con v erges to a random v ariable X ′ whic h is finite a.s.. Now b y the Birk off strong law of large n um bers for ergo dic sequences, a.s. lim N →∞ 1 N 0 X i = − N +1 Z i = E Z 1 < 0 (cf. Theorem 3.5.7. in Stout [10]), hence a.s. lim N →∞ 0 X i = − N +1 Z i = −∞ . 2 W e go t that there is a random v ariable τ suc h that for all i > 0 ∞ > X − τ , 0 = X − τ − i, 0 , and therefore X ′ = sup n ≥ 0 V n . Step 2. P ut X ′ 0 = X ′ and fo r n ≥ 0, X ′ n +1 = ( X ′ n + Z n +1 ) + . W e show that { X ′ i } is statio nary and ergo dic. F or a n y sequence z ∞ −∞ = ( . . . , z − 1 , z 0 , z 1 , . . . ) put F ( z ∞ −∞ ) = lim N →∞ max { 0 , z 0 , z 0 + z − 1 , . . . , z 0 + . . . + z − N } . Then by Step 1 X ′ 0 = X ′ = F ( Z ∞ −∞ ) . W e prov e by induction tha t for n ≥ 0, X ′ n = F ( T n Z ∞ −∞ ) , where T is the left shift. F or n = 1, F ( T Z ∞ −∞ ) = lim N →∞ max { 0 , Z 1 , Z 1 + Z 0 , . . . , Z 1 + Z 0 + . . . , Z − N +1 } = ( lim N →∞ max { 0 , max { 0 , Z 0 , . . . , Z 0 + . . . , Z − N +2 } + Z 1 } ) = (max { 0 , [ lim N →∞ max { 0 , Z 0 , . . . , Z 0 + . . . , Z − N +2 } ] + Z 1 } ) = ( X ′ 0 + Z 1 ) + = X ′ 1 . No w w e prov e from n to n + 1. X ′ n +1 = ( X ′ n + Z n +1 ) + = ( F ( T n Z ∞ −∞ ) + Z n +1 ) + = ( lim N →∞ max { 0 , Z n , Z n + Z n − 1 , . . . , Z n + Z n − 1 + . . . , Z n − N } + Z n +1 ) + = lim N →∞ max { 0 , Z n +1 , Z n +1 + Z n , . . . , Z n +1 + Z n + . . . , Z n +1 − N } = F ( T n +1 Z ∞ −∞ ) . Step 3. S imilarly to the pro of of Step 1, X n = max { 0 , Z n , Z n + Z n − 1 , . . . , Z n + . . . + Z 1 , Z n + . . . + Z 1 + X 0 } , 3 and X ′ n = max { 0 , Z n , Z n + Z n − 1 , . . . , Z n + . . . + Z 1 , Z n + . . . + Z 1 + X ′ 0 } . But for large n , b oth Z n + . . . + Z 1 + X 0 < 0 and Z n + . . . + Z 1 + X ′ 0 < 0 , and so X n = X ′ n = max { 0 , Z n , Z n + Z n − 1 , . . . , Z n + . . . + Z 1 } . The pro o f of Theorem 1 is complete. Remark 1. F rom the pro of it is clear that the fo llo wing extension is straightforw ard: If { Z i } is coupled with a statio nary and ergodic { Z ′ i } , and E { Z ′ i } < 0, then { X i } is coupled with a statio nary and ergo dic { X ′ i } suc h that X ′ 0 = sup n ≥ 0 V n , where V 0 = 0 , V n = n − 1 X i =0 Z ′ − i , ( n ≥ 1) . 2.2 Queue l ength for discrete time queueing As an application of Theorem 1 cons ider a discrete time qu eueing with constant service rate s , and denote b y Y n the num b er of arriv als in time slot n . L et the initial length of the queue Q 0 b e arbitrary non-negative in teger v alued random v a riable. Then Q n +1 = ( Q n − s + Y n +1 ) + for n ≥ 0. Put V 0 = 0 , V n = n − 1 X i =0 Y − i − ns, ( n ≥ 1) . Corollary 1 If { Y i } is stationary and er go di c , and E { Y i } < s , then then { Q i } is c ouple d with a s tationary and er go d ic { Q ′ i } such that Q ′ 0 = sup n ≥ 0 V n . 4 Pro of. Apply Theorem 1 f or Z n = Y n − s. Remark 2. This result together w ith Remark 1 has some consequences for net w ork of serv ers (tandem of queue s), when the output o f a serv er ( Q n + Y n +1 − Q n +1 ) is the input of a nother serv er. It is easy to sho w that the output ( Q n + Y n +1 − Q n +1 ) is coupled with a stationary and ergo dic sequence, and the exp ectations of the input and the output are equal, so the stabilit y condition holds for the next serv er, t o o. 2.3 W ating ti me for generalized G/G/1 This is another application of Theorem 1. According to L indley (1952 ) consider the extension of the G/G/1 model. Let W n b e the waiting time of the n - th arriv al, S n b e the service time of the n -th arriv al, and T n +1 b e the inter arr iv al time b et w een the ( n + 1)-th and n -th arriv als. Let W 0 b e an ar bitrary random v ariable. Then W n +1 = ( W n − T n +1 + S n ) + for n ≥ 0. Put V 0 = 0 , V n = n − 1 X i =0 ( S − i − 1 − T − i ) , ( n ≥ 1) . Corollary 2 (Extension of L oynes (1 9 64)) If { S i − 1 − T i } is stationary and er go dic, E { S i − 1 } < E { T i } , then { W i } is c ouple d with a stationary and er g o dic { W ′ i } such that W ′ 0 = sup n ≥ 0 V n . Pro of. Apply Theorem 1 f or Z n = S n − 1 − T n . 3 Limit dis tribution In a queueing problem the prop erties of the limit distribution are of great imp ort a nce. In this section we consider the sp ecial case of Section 2.2., when the a r riv als { Y n } are ergo dic and the service rate is constan t s . If { Y n } are w eakly dep enden t then the ta il of the limit distribution is almost exponen tial, whic h ma y result in efficien t algorithms for call admission con trol (cf. Duffield, Lewis, O’Connel, Russel T o o mey (19 95)). The exponential tail distribution can b e deriv ed using large deviation tec hnique (cf. Glynn, Whitt (19 9 4)). The basic to ol in this resp ect is the cumm ulan t moment generating function: λ ( θ ) = lim n →∞ 1 n log E { e θ P n k =1 Y k } , 5 assuming tha t this limit exists. If the set { θ ; λ ( θ ) − θ s < 0 } , is not empt y then put δ = sup { θ ; λ ( θ ) − θ s < 0 } . Then for large q P { Q > q } ≃ e − δq . The question is whether under the stabilit y condition E ( Y 1 ) < s one has exp onen tial tail distribution. Glynn, Whitt (1 994) ga v e a p ositive answ er under w eakly dep enden t { Y n } . The limit in the definition of λ ( θ ) exists only under some conditions, for example, if { Y n } fo rm a binary Mark o v chain then λ ( θ ) can b e calculated (D em b o, Zeitouni (1992 )), an explicite b o und on P { Q > q } can b e g iv en (Duffield ( 1 994)). F or a p ossible extension the other pro blem is that E ( Y 1 ) < s is only a necessary condition that the set { θ ; λ ( θ ) − θ s < 0 } is no t empt y , but not sufficien t. This can b e seen by Jensen’s inequalit y: 1 n log E { e θ P n k =1 Y k } > 1 n log e θ E { P n k =1 Y k } = θ E { Y 1 } , therefore if the set { θ ; λ ( θ ) − θ s < 0 } is not empty then E ( Y 1 ) < s . F or long range dep enden t arriv als Duffield, O’Connel (199 5 ) pro v ed that t he ta il may not b e exp onen tial. They introduced the scaled cumm ulan t momen t generating function: λ ∗ ( θ ) = lim n →∞ 1 v ( n ) log E { e θ v ( n ) a ( n ) [ P n k =1 Y k − ns ] } , assuming that this limit exists, where a ( t ) and v ( t ) are monoton increasing functions. If the set { θ ; λ ∗ ( θ ) < 0 } , is not empt y then put δ = sup { θ ; λ ∗ ( θ ) < 0 } . Then for large q P { Q > q } ≃ e − δv ( a − 1 ( q )) . Duffield, O’Connel (1995) applied this result when { P n k =1 Y k } is a Ga ussian pro cess with stationary incremen ts, or Ornstein-Uhlen beck pro cess , or a squared Bessel pro cess . These examples can b e motiv ated b y mu ltiplexing many sources. F o r all these examples Y n is un bo unded. In this section w e consider b ounded Y n , esp ecially binar y v alued Y n . Sho w examples such that Q has la rger tail than exp onen tial. Prop osition 1 Ther e is a stationary, er go dic an d binary value d { Y n } such that E { Y 1 } ≤ 1 / 2 and with s = 3 / 4 the queue length se quenc e is stable, and for e ach δ > 0 ther e is q such that P { Q > q } > e − δq . 6 Pro of. F rom Theorem 1 Q = sup n ≥ 0 V n , therefore for an y n P { Q > q } ≥ P { V n > q } = P { n − 1 X j =0 Y − j > ns + q } . W e show that for δ i = 1 i , q i = 2 i 2 , n i = 2 i − 1 ( i > 16), P ( n i − 1 X j =0 Y − j > 3 4 n i + q i ) > 2 − δ i q i . Step 1. W e presen t first a dynamical system giv en in Gy¨ orfi, Morv ai, Y ak o witz (1998). W e will define a transformation T on the unit in terv al. Consider the binary expansion r ∞ 1 of eac h real-n um b er r ∈ [0 , 1), that is, r = P ∞ i =1 r i 2 − i . When there are tw o expansions, use the represen tatio n whic h con tains finitely many 1 ′ s . No w let τ ( r ) = min { i > 0 : r i = 1 } . (3) Notice that, aside from the exce ptional set { 0 } , whic h has Leb esgue measure zero τ is finite and we ll-defined o n the closed unit in terv al. The transformation is defined by ( T r ) i =      1 if 0 < i < τ ( r ) 0 if i = τ ( r ) r i if i > τ ( r ) . (4) Step 2. W e show that the transformatio n T is ergo dic. Notice that in fact, T r = r − 2 − τ ( r ) + P τ ( r ) − 1 l =1 2 − l . All iterat io ns T k of T for −∞ < k < ∞ are we ll defined a nd in v ertible with the exeption of the set of dy adic rationals whic h has Leb esgue measure zero. In the futur e w e will neglec t this set. T ransformation T could b e defined recursiv ely as T r = ( r − 0 . 5 if 0 . 5 ≤ r < 1 1+ T ( 2 r ) 2 if 0 ≤ r < 0 . 5. Let S i = { I i 0 , . . . , I i 2 i − 1 } b e a pa rtition of [0 , 1) where for eac h in teger j in the ra nge 0 ≤ j < 2 i I i j is defined as the set of n um b ers r = P ∞ v =1 r v 2 − v whose binary expansion 0 .r 1 , r 2 , . . . star t s with the bit sequence j 1 , j 2 , . . . , j i that is rev ersing the binar y expans ion j i , . . . , j 2 , j 1 of the num b er j = P i l =1 2 l − 1 j l . Observ e that in S i there are 2 i left-semiclosed in terv als and eac h in terv al I i j has length (Leb esgue measure) 2 − i . No w I i j is mapp ed linearly , under T on to I i j − 1 for 7 j = 1 , . . . , 2 i − 1. T o confirm this, observ e that fo r j = 1 , . . . , 2 i − 1, if r ∈ I i j then T r = τ ( r ) − 1 X l =1 2 − l + ∞ X l = τ ( r )+1 r l 2 − l = r − i X l =1 2 − l ( j l − ( j − 1) l ) = i X l =1 ( j − 1) l 2 − l + ∞ X l = i +1 r l 2 − l . No w if 0 < r ∈ I i 0 then τ ( r ) > i and so T r ∈ I i 2 i − 1 . F urthermore, if r ∈ I i 2 i − 1 then r 1 = . . . = r i = 1, and th us conclude that ( T − 1 r ) 1 = . . . = ( T − 1 r ) i = 0, that is, T − 1 r ∈ I i 0 . Let r ∈ [0 , 1) and n ≥ 1 b e arbitrary . Then r ∈ I n j for some 0 ≤ j ≤ 2 n − 1 . F or all j − (2 n − 1) ≤ k ≤ j , T k r = n X l =1 ( j − k ) l 2 − l + ∞ X l = n +1 r l 2 − l . (5) No w since T − 1 I i j = I i j +1 for i ≥ 1, j = 0 , . . . , 2 i − 2, a nd the union o v er i and j of these sets generate the Borel σ -a lg ebra, w e conclude that T is measurable. Similar reasoning shows that T − 1 is also measurable . The dynamical system (Ω , F , µ, T ) is iden tified with Ω = [0 , 1) and F the Borel σ -a lgebra on [0 , 1), T b eing the tra nsfor ma t io n dev elop ed abov e. T ake µ to b e Leb esgue measure on the unit in terv al. Since tr a nsformation T is measure-preserving on eac h set in the collection { I i j : 1 ≤ j ≤ 2 i − 1 , 1 ≤ i < ∞} and these interv als generate the Borel σ -algebra F , T is a stationary transformation. Now w e pro v e that transformation T is ergo dic as we ll. Assume T A = A . If r ∈ A then T l r ∈ A for −∞ < l < ∞ . Let R n : [0 , 1) → { 0 , 1 } b e the function R n ( r ) = r n . If r is c hosen uniformly on [0 , 1) then R 1 , R 2 , . . . is a series if i.i.d. random v ariables. Let F n = σ ( R n , R n +1 , . . . ). By (5) it is immediate that A ∈ ∩ ∞ n =1 F n and so A is a tail eve n t. By Kolmogorov’s zero one la w µ ( A ) is either zero or one. Hence T is ergo dic. Step 3. W e define a pa r tition of [0 , 1) in the f o llo wing w a y . Let A 0 = ∅ , B 0 = [0 , 2 − 2 ), C 0 = A 0 S B 0 . In g eneral, for i ≥ 1 let A i = 2 i − 1 − 1 [ j =0 T − j [0 , 2 − 2 i − 1 ) = 2 i − 1 − 1 [ j =0 I 2 i +1 j (6) and B i = 2 i − 1 [ j =2 i − 1 T − j [0 , 2 − 2 i − 1 ) = 2 i − 1 [ j =2 i − 1 I 2 i +1 j . (7) W e show that µ ( A i ) = µ ( B i ) = 2 − i − 2 . Since [0 , 2 − 2 i − 1 ) = I 2 i +1 0 it is clear for 0 ≤ j < 2 i the sets T − j [0 , 2 − 2 i − 1 ) ar e disjoin t. Thus µ ( A i ) = µ ( B i ) = 2 − 2 i − 1 2 i − 1 . 8 Step 4. Put C i = A i [ B i . (8) and C = ∞ [ j =0 C i . W e show that µ ( C ) ≤ 1 2 . Since A 0 = ∅ and for i ≥ 1, A i = 2 i − 1 − 1 [ j =0 T − j [0 , 2 − 2 i − 1 ) =   2 i − 1 − 1 − 1 [ j =0 T − j [0 , 2 − 2 i − 1 )   [   2 i − 1 − 1 [ j =2 i − 1 − 1 T j [0 , 2 − 2 i − 1 )   ⊆   2 i − 1 − 1 − 1 [ j =0 T − j [0 , 2 − 2( i − 1) − 1 )   [   2 i − 1 − 1 [ j =2 i − 1 − 1 T j [0 , 2 − 2( i − 1) − 1 )   = A i − 1 [ B i − 1 = C i − 1 and so A i ⊆ C i − 1 , that is, C = ∞ [ i =0 B i . F urthermore µ ( C ) ≤ ∞ X i =0 µ ( B i ) = ∞ X i =0 2 − 2 i − 1 2 i − 1 = ∞ X i =0 2 − i − 2 = 1 2 . Step 5. Define the binary t ime series { Y i } as Y i ( ω ) = ( 1 if T i ω ∈ C 0 otherw ise. Clearly { Y i } is stationary and ergo dic since the dynamical system itself w as so. E Y i ≤ 0 . 5 since µ ( C ) ≤ 0 . 5. If ω ∈ A i then by (6), (7) and (8 ) ω , T − 1 ω , . . . , T − (2 i − 1 − 1) ω ∈ C 9 that is Y 0 ( ω ) = 1 , . . . , Y − ( n i − 1) ( ω ) = 1 . F urthermore for i > 16, n i 4 = 2 i − 1 4 > 8 i 2 4 = q i and so n i = 3 4 n i + 1 4 n i > 3 4 n i + q i . Th us P ( n i − 1 X j =0 Y − j > 3 4 n i + q i ) ≥ µ ( A i ) . By Step 3, f o r i > 1 6, µ ( A i ) = 2 − i − 2 > 2 − 2 i = 2 − (1 /i )2 i 2 = 2 − δ i q i . The pro o f of Prop osition 1 is complete. One could define λ ( θ ) := lim sup n →∞ 1 n log E e θ P n i =1 Y i . The next p op osition sho ws that for the stationa ry and ergo dic time-series just defined, { θ ; λ ( θ ) − θ s < 0 } is the em t y set when s = 1. Prop osition 2 F or { Y i } defi n e d in the pr o of of Pr op osition 1, lim i →∞ 1 n i log E e P n i − 1 j =0 θ Y j = θ . Pro of. By Step 3 a nd 5 o f the pro of of Prop osition 1 θ ≥ lim sup i →∞ 1 n i log E e P n i − 1 j =0 θ Y j ≥ lim inf i →∞ 1 n i log 2 e log 2 E 2 P n i − 1 j =0 (log 2 e ) θY j ≥ lim inf i →∞ 1 n i log 2 e log 2 (2 (log 2 e ) θn i µ ( A i )) = lim inf i →∞ θ + 1 n i log 2 e log 2 µ ( A i ) = θ + lim i →∞ 2 − i +1 log 2 e log 2 − i − 2 = θ. The pro o f of Prop osition 2 is complete. 10 References [1] Dem b o, A. a nd Zeitouni, O. (1992). L ar ge Deviations T e chn i q ues and Applic ations . Jones and Bartlett Publishers. [2] N. G . Duffield (1994) ”Exp onen tial b ounds for queues with Mark o vian arriv als”, Queue- ing Systems , 17 , 413-430. [3] N. G. Duffield, J. T. Lewis, N. O’Connel, R. Russel, F. F o omey . (1995) ”Entrop y of A TM traffic streams: a to o l for estimating QoS parameters” IEEE J. Sele cte d Ar e as in Communic a tions , 13 , 981- 9 89. [4] N. G. Duffield, N. O’Connel (1995 ) ”Large deviations a nd ov erflow probabilities for the general single-serv er queue, with applications”, Pr o c. Cam bridge Philos. So c. , 118, 363-374 . [5] P . W. Glynn, W. Whitt (199 4) ”Logar it hmic asymptotics for steady-state tail proba- bilities in a single-serv er queue”, J. Applie d Pr o b ability , 118, 3 63-374. [6] L. Gy¨ orfi, G. Morv ai, S. Y ak o witz (1998) ”Limits to consisten t on-line forecasting for ergo dic time series”, I EEE T r ans. Information The ory , 44, 886 -892. [7] D. V. Lindley (195 2 ) ” The theory of queues with a single serv er” Pr o c. Cambridge Philos. So c. , 48, 277-2 89. [8] T. Lindv all. (1 992) L e ctur es on the C oupling Metho d . Wiley , New Y o rk. [9] R. M. Loy nes. (1962) ”The stability of a queue with non-indep enden t in ter-arriv al and service times”, Pr o c. Cam bridge Philos. So c. , 58, 497- 520. [10] W.F. Stout. (19 74) A lmost sur e c on ver genc e . Academic Press, New Y ork. 11