Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights Maaike Verloop (CWI) Urtzi Ayesta (CNRS-LAAS & BECAM) Sindo N´ u˜ nez-Queija (University of Amsterdam & CWI)
Heavy-traffic analysis of the M/PH/1 discrimina- tory processor sharing queue with phase-dependent weights • Discriminatory Processor Sharing vs Egalitarian processor Sharing • Dynamics in heavy traffic • Proof for phase type distributions R. N´ u˜ nez-Queija 1
Model description Class k • Poisson arrivals, λ k = Λ p k • Service B k ( x ) := P ( B k ≤ x ) • Load ρ k = λ k β k • # customers N k w k • Service rate � K j =1 N j w j Stability ρ = � K k =1 ρ k < 1 R. N´ u˜ nez-Queija 2
Model description Class k • Poisson arrivals, λ k = Λ p k • Service B k ( x ) := P ( B k ≤ x ) • Load ρ k = λ k β k • # customers N k w k • Service rate � K j =1 N j w j Stability ρ = � K k =1 ρ k < 1 R. N´ u˜ nez-Queija 2-1
Model description Class k • Poisson arrivals, λ k = Λ p k • Service B k ( x ) := P ( B k ≤ x ) • Load ρ k = λ k β k • # customers N k w k • Service rate � K j =1 N j w j Stability ρ = � K k =1 ρ k < 1 R. N´ u˜ nez-Queija 2-2
Model description Class k • Poisson arrivals, λ k = Λ p k • Service B k ( x ) := P ( B k ≤ x ) • Load ρ k = λ k β k • # customers N k w k • Service rate � K j =1 N j w j Stability ρ = � K k =1 ρ k < 1 R. N´ u˜ nez-Queija 2-3
Model description Class k • Poisson arrivals, λ k = Λ p k • Service B k ( x ) := P ( B k ≤ x ) • Load ρ k = λ k β k • # customers N k w k • Service rate � K j =1 N j w j Stability ρ = � K k =1 ρ k < 1 R. N´ u˜ nez-Queija 2-4
Model description Class k • Poisson arrivals, λ k = Λ p k • Service B k ( x ) := P ( B k ≤ x ) • Load ρ k = λ k β k • # customers N k w k • Service rate � K j =1 N j w j Stability ρ = � K k =1 ρ k < 1 Applications: time-shared computing systems, TCP, ADSL R. N´ u˜ nez-Queija 2-5
Literature • Kleinrock [1967] (‘Priority Processor-shared Model’) • Fayolle, Mitrani & Iasnogorodski [1980] • Grishechkin [1992, 1994] • Rege & Sengupta [1994, 1996] • Borst, Van Ooteghem & Zwart [2003] • Altman, Jimenez & Kofman [2004] • Avrachenkov, Ayesta, Brown, N-Q [2005] R. N´ u˜ nez-Queija 3
Discriminatory PS versus (Egalitarian) PS PS: Mean queue lengths are entirely “insensitive” as op- posed to non-preemptive disciples like FCFS ρ k E N k = 1 − ρ For DPS: E N k are all finite under the usual stability condition (regardless of the higher-order moments of the service re- quirements) Other insensitivity properties of PS only carry over to DPS in asymptotic regimes R. N´ u˜ nez-Queija 4
Discriminatory PS versus Egalitarian PS PS: Expected sojourn time for jobs of given size E T k ( x ) 1 = 1 − ρ x DPS: true in the limit as x → ∞ [Fayolle, Mitrani & Iasno- gorodski] and the ”bias” is also insensitive j λ j (1 − w k w j ) E (( B j ) 2 ) � � � x lim E T k ( x ) − = . 2(1 − ρ ) 2 x →∞ 1 − ρ R. N´ u˜ nez-Queija 4-1
Discriminatory PS versus Egalitarian PS Tails of the sojourn time distributions PS and DPS: For regularly varying service requirement distributions with finite variance (conditions can be relaxed): P { T k > x } P { B k > (1 − ρ ) x } → 1 , as x → ∞ Again, the “scaling factor” 1 − ρ is insensitive and common to all classes R. N´ u˜ nez-Queija 4-2
Discriminatory PS versus Egalitarian PS Time-scale separation (1): Class k operates on a much faster time scale than class k + 1 , for all k = 1 , 2 , . . . , K − 1 • arrival rates λ k f k ( r ) • service requirements of class k distributed as B k /f k ( r ) • with f k +1 ( r ) /f k ( r ) → 0 as r → ∞ R. N´ u˜ nez-Queija 4-3
Discriminatory PS versus Egalitarian PS Time-scale separation (2): The limiting distribution of the slow class is geometric ρ 2 ρ 2 ) n 2 P r { N 2 = n 2 } → (1 − )( 1 − ρ 1 1 − ρ 1 and P r { N 1 = n 1 | N 2 = n 2 } Γ( n 1 + n 2 w 2 + 1) n 2 w 2 w 1 +1 w 1 + 1) ρ n 1 → 1 (1 − ρ 1 ) Γ( n 1 + 1)Γ( n 2 w 2 w 1 For PS all limits can be replaced with equalities R. N´ u˜ nez-Queija 4-4
Dynamics R. N´ u˜ nez-Queija 5
Dynamics R. N´ u˜ nez-Queija 5-1
Dynamics R. N´ u˜ nez-Queija 5-2
Dynamics R. N´ u˜ nez-Queija 5-3
Dynamics (2) R. N´ u˜ nez-Queija 6
Dynamics (2) R. N´ u˜ nez-Queija 6-1
Heavy traffic: Theorem For phase-type distributions → E · ( ¯ ρ 1 , ¯ ρ 2 , . . . , ¯ ρ K (1 − ρ )( N 1 , N 2 , . . . , N K ) d ) w 1 w 2 w K where E is exponential with mean k p k E [( B k ) 2 ] / � k p k E B k � k 1 ρ k E [( B k ) 2 ] / E B k w k ¯ � State-space collapse: “In heavy traffic (1 − ρ ) N 1 , . . . , (1 − ρ ) N K are proportional to a common exponentially distributed random variable” R. N´ u˜ nez-Queija 7
Heavy traffic: Theorem For phase-type distributions → E · ( ¯ ρ 1 , ¯ ρ 2 , . . . , ¯ ρ K (1 − ρ )( N 1 , N 2 , . . . , N K ) d ) w 1 w 2 w K where E is exponential with mean k p k E [( B k ) 2 ] / � k p k E B k � k 1 ρ k E [( B k ) 2 ] / E B k w k ¯ � State-space collapse: “In heavy traffic (1 − ρ ) N 1 , . . . , (1 − ρ ) N K are proportional to a common exponentially distributed random variable” R. N´ u˜ nez-Queija 7-1
Heavy traffic: Interpretation → E · ( ¯ , ¯ , . . . , ¯ ρ 1 ρ 2 ρ K (1 − ρ )( N 1 , N 2 , . . . , N K ) d ) w 1 w 2 w K where E is exponential with mean k p k E [( B k ) 2 ] / � � k p k E B k k 1 ρ k E [( B k ) 2 ] / E B k w k ¯ � Assume that N i /N j = n i /n j for some constants n k (as in the exponential case), then J w j n j � = µ p 0 i a ij � J i =1 w i n i i =1 normalizing � J i =1 w i n i = 1 gives the result up to a multi- plicative factor R. N´ u˜ nez-Queija 8
Heavy traffic: Interpretation → E · ( ¯ ρ 1 , ¯ ρ 2 , . . . , ¯ ρ K (1 − ρ )( N 1 , N 2 , . . . , N K ) d ) w 1 w 2 w K where E is exponential with mean k p k E [( B k ) 2 ] / � � k p k E B k k 1 ρ k E [( B k ) 2 ] / E B k w k ¯ � Assume that N ′ i /N ′ j = n i /n j for some constants n k (as in the exponential case), then J w j n j � = µ p 0 i a ij � J i =1 w i n i i =1 normalizing � J i =1 w i n i = 1 gives the result up to a multi- plicative factor Work-conserving and non-idling: ρ p k E [( B k ) 2 ] / � � E V = p k E B k 2(1 − ρ ) k k R. N´ u˜ nez-Queija 8-1
Heavy traffic: Interpretation → E · ( ¯ ρ 1 , ¯ ρ 2 , . . . , ¯ ρ K (1 − ρ )( N 1 , N 2 , . . . , N K ) d ) w 1 w 2 w K where E is exponential with mean k p k E [( B k ) 2 ] / � � k p k E B k k 1 ρ k E [( B k ) 2 ] / E B k w k ¯ � Assume that N ′ i /N ′ j = n i /n j for some constants n k (as in the exponential case), then J w j n j � = µ p 0 i a ij � J i =1 w i n i i =1 normalizing � J i =1 w i n i = 1 gives the result up to a multi- plicative factor Work-conserving and non-idling: ρ p k E [( B k ) 2 ] / � � E V = p k E B k 2(1 − ρ ) k k R. N´ u˜ nez-Queija 8-2
Phase-type service requirements (1) Recall • Poisson arrivals, λ k = Λ p k • Service B k ( x ) := P ( B k ≤ x ) • Load ρ k = λ k β k • # customers N k w k • Service rate � K j =1 N j w j Stability ρ = � K k =1 ρ k < 1 R. N´ u˜ nez-Queija 9
Phase-type service requirements (2) Class k has m k service phases total # service phases: � K k =1 m k := J p 0 i = probability that arriving customer is in phase i µ i = service rate in phase i g i = service weight in phase i p ij = phase transition probabilities p i 0 = probability of completing service after phase i n i g i n � = ¯ g i (¯ n ) := , if ¯ 0 � K k =1 n k g k 10
Phase-type service requirements (2) Class k has m k service phases total # service phases: � K k =1 m k := J p 0 i = probability that arriving customer is in phase i µ i = service rate in phase i g i = service weight in phase i p ij = phase transition probabilities p i 0 = probability of completing service after phase i n i g i n � = ¯ g i (¯ n ) := , if ¯ 0 � K k =1 n k g k 10-1
Phase-type service requirements (2) Class k has m k service phases total # service phases: � K k =1 m k := J p 0 i = probability that arriving customer is in phase i µ i = service rate in phase i g i = service weight in phase i p ij = phase transition probabilities p i 0 = probability of completing service after phase i n i g i n � = ¯ g i (¯ n ) := , if ¯ 0 � K k =1 n k g k 10-2
Phase-type service requirements (2) Class k has m k service phases total # service phases: � K k =1 m k := J p 0 i = probability that arriving customer is in phase i µ i = service rate in phase i g i = service weight in phase i p ij = phase transition probabilities p i 0 = probability of completing service after phase i n i g i n � = ¯ g i (¯ n ) := , if ¯ 0 � K k =1 n k g k 10-3
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