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

Youri Raaijmakers (), Sem Borst and Onno Boxma
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Youri Raaijmakers: Eindhoven University of Technology
Sem Borst: Eindhoven University of Technology
Onno Boxma: Eindhoven University of Technology

Queueing Systems: Theory and Applications, 2023, vol. 103, issue 1, No 4, 159 pages

Abstract: Abstract 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 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.

Keywords: Parallel-server systems; Fork–join; Redundancy; Heavy-tailed distributions; Response time asymptotics; Primary 60K25; 90B22; Secondary 60F10 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s11134-022-09856-6

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