Stability and tail behavior of redundancy systems with processor sharing

Raaijmakers, Youri; Borst, Sem; Boxma, Onno

doi:10.1016/j.peva.2021.102195

Cited by 3 publications

(3 citation statements)

References 24 publications

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“…−dν [26] the tail index is independent of the load of the system and for identical replicas even independent of the number of replicas. In the special case of redundancy-d scheduling with identical and i.i.d.…”

Section: Redundancy Schedulingmentioning

confidence: 99%

“…variant the LCFS-PR discipline has better tail asymptotics than the FCFS discipline for scenarios with low load and a small number of replicas; in all other scenarios both service disciplines have similar tail asymptotics. In [26] it is shown that for the c.o.c. variant of redundancy-d scheduling with the PS discipline the tail index of the response time is −ν for identical replicas and −dν for i.i.d.…”

Section: Redundancy Schedulingmentioning

confidence: 99%

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Fork-join and redundancy systems with heavy-tailed job sizes

Raaijmakers¹,

Borst²,

Boxma³

2021

Preprint

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We investigate the tail asymptotics of the response time distribution for the cancel-on-start (c.o.s.) and cancel-on-completion (c.o.c.) variants of redundancyd scheduling and the fork-join model with heavy-tailed job sizes. We present bounds, which only differ in the pre-factor, for the tail probability of the response time in the case of the first-come first-served (FCFS) discipline. For the c.o.s. variant we restrict ourselves to redundancy-d scheduling, which is a special case of the fork-join model. In particular, for regularly varying job sizes with tail index −ν the tail index of the response time for the c.o.s. variant of redundancyd equals − min{d cap (ν − 1), ν}, where d cap = min{d, N − k}, N is the number of servers and k is the integer part of the load. This result indicates that for d cap < ν ν−1 the waiting time component is dominant, whereas for d cap > ν ν−1 the job size component is dominant. Thus, having d = ⌈min{ ν ν−1 , N − k}⌉ replicas is sufficient to achieve the optimal asymptotic tail behavior of the response time. For the c.o.c. variant of the fork-join(n F , n J ) model the tail index of the response time, under some assumptions on the load, equals 1 − ν and 1 − (n F + 1 − n J )ν, for identical and i.i.d. replicas, respectively; here the waiting time component is always dominant.

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Section: Redundancy Schedulingmentioning

confidence: 99%

Section: Redundancy Schedulingmentioning

confidence: 99%

Fork-join and redundancy systems with heavy-tailed job sizes

Raaijmakers¹,

Borst²,

Boxma³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the FCFS discipline we have that if X min ∈ RV (−ν) then R ∈ ORV (1−ν) (see [10]), whereas for the PS discipline we have that if X min ∈ RV (−ν) then R ∈ ORV (−ν) (see [11]). These results indicate that for heavy-tailed job size distributions the PS discipline always has better tail behavior than the FCFS discipline for all dependency structures between the replicas.…”

Section: Tail Behaviormentioning

confidence: 99%

Comparison of the FCFS and PS discipline in Redundancy Systems

Raaijmakers¹

2021

Preprint

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We consider the c.o.c. redundancy system with N parallel servers where incoming jobs are immediately replicated to d servers chosen uniformly at random (without replacement). A job finishes service as soon as the first replica is completed, after which all the remaining replicas are abandoned. We compare the performance of the first-come first-served (FCFS) and processor-sharing (PS) discipline based on the stability condition, the tail behavior of the latency and the expected latency.

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Fork–join and redundancy systems with heavy-tailed job sizes

2022

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We investigate the tail asymptotics of the response time distribution for the cancel-on-start (c.o.s.) and cancel-on-completion (c.o.c.) variants of redundancy-d scheduling and the fork–join model with heavy-tailed job sizes. We present bounds, which only differ in the pre-factor, for the tail probability of the response time in the case of the first-come first-served discipline. For the c.o.s. variant, we restrict ourselves to redundancy-d scheduling, which is a special case of the fork–join model. In particular, for regularly varying job sizes with tail index-$$\nu $$ ν the tail index of the response time for the c.o.s. variant of redundancy-d equals -$$\min \{d_{\mathrm {cap}}(\nu -1),\nu \}$$ min { d cap ( ν - 1 ) , ν } , where $$d_{\mathrm {cap}} = \min \{d,N-k\}$$ d cap = min { d , N - k } , N is the number of servers and k is the integer part of the load. This result indicates that for $$d_{\mathrm {cap}} < \frac{\nu }{\nu -1}$$ d cap < ν ν - 1 the waiting time component is dominant, whereas for $$d_{\mathrm {cap}} > \frac{\nu }{\nu -1}$$ d cap > ν ν - 1 the job size component is dominant. Thus, having $$d = \lceil \min \{\frac{\nu }{\nu -1},N-k\} \rceil $$ d = ⌈ min { ν ν - 1 , N - k } ⌉ replicas is sufficient to achieve the optimal asymptotic tail behavior of the response time. For the c.o.c. variant of the fork–join ($$n_{\mathrm {F}},n_{\mathrm {J}}$$ n F , n J ) model, the tail index of the response time, under some assumptions on the load, equals $$1-\nu $$ 1 - ν and $$1-(n_{\mathrm {F}}+1-n_{\mathrm {J}})\nu $$ 1 - ( n F + 1 - n J ) ν , for identical and i.i.d. replicas, respectively; here, the waiting time component is always dominant.

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Stability and tail behavior of redundancy systems with processor sharing

Cited by 3 publications

References 24 publications

Fork-join and redundancy systems with heavy-tailed job sizes

Fork-join and redundancy systems with heavy-tailed job sizes

Comparison of the FCFS and PS discipline in Redundancy Systems

Fork–join and redundancy systems with heavy-tailed job sizes

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