Here, we consider a system that randomly maps a job toH out ofK available servers based on a uniform distribution over the set{A⊆ {1, . . . , K} | |A|=H}
of server combinations with cardinality H. We bound the job waiting and re- sponse time in this system using the following abstraction which considers the probability of assigning a task to a specific server. Note that the probability for a task dedicated to a certain server is given bypd=H/K. Now, if we focus on
only one server of this FJ system, the task service times at that server can be represented by the compound distribution
¯ xk,i=
xk,i with probability pd
0 with probability 1−pd ,
(6.36)
since a job that is not assigned to this server can be considered to have a ser- vice time equal to 0. Hence, one server of this FJ system with random partial mapping can be modelled as if it is part of a FJ system with full mapping as in
Section 6.2, but with the modified service times ¯xk,i. Note that due to the selec-
tion of the subset with fixed cardinalityH, the (¯xk,i)kare no longer independent.
Their MGF can be computed as:
Eeθ¯xk,i= (1−p
d) +pdEeθxk,i .
The representations for the waiting and response time, respectively, be- come w=D max n≥0 ( max 1≤k≤H ( n X i=1 ¯ xk,i− n X i=1 ti )) , (6.37) and r=D max n≥0 ( max 1≤k≤H ( xk,0+ n X i=1 ¯ xk,i− n X i=1 ti )) . (6.38) Note the asymmetry for the response time in (6.38). For i ≥ 1 we consider the modified service times ¯xk,i as the corresponding server is only selected with
probability pd. In turn, for i = 0, we need to consider the unmodified service
timex0,ias we only look at those servers which have been selected for mapping.
The following theorems provide upper bounds on the steady-state wait- ing and response time distributions in the non-blocking scenarios with partial random mapping for renewal and Markov-modulated interarrivals, respectively.
Theorem 6.8. (Random Mapping, Renewals, Non-Blocking) Given a FJ queueing system with K servers and random partial mapping of jobs toH ≤K
servers based on a uniform distribution over the set{A⊆ {1, . . . , K} | |A|=H}
of server combinations with cardinality H. The system is fed with renewal job arrivals. If the task service times xk,j are i.i.d., then the steady-state waiting and response timeswandr are bounded by
P[w≥σ]≤He−θ∗σ ,
waiting time probability λ =0.9 λ =0.75 λ =0.5 0 10 20 30 40 50 10 −6 10 −4 10 −2 10 0 (a) Impact ofλ waiting time probability m N=0.75 m N=0.5 m N=0.25 0 10 20 30 40 50 10 −6 10 −4 10 −2 10 0
(b) Impact of the fan-out ratioH/N
Figure 6.8: Bounds on the waiting time distributions vs. simulation box-plots for renewal input with random server mapping. The parameters are K = 16, µ = 1. (a) Here, we fix the fan-out ratio to H = 12 and change the job arrival rate λ ∈ {0.5,0.75,0.9} while in (b) we fix the arrival rate to λ= 0.75 and vary the fan-out ratio H/K ∈
{0.25,0.5,0.75}. Simulations include 100 runs, each accounting for 106 slots.
whereθ is the solution of
θ∗:=
θ >0
(1−pd) +pdE
eθxn,iEe−θt1= 1 . (6.39)
Proof. The proof goes along similar steps as for Theorem 6.7, however, using the process
zk(n) =eθ
∗Pn
i=1(¯xk,i−ti)
which is a martingale for eachk≤Kunder the criterion (6.39) onθ∗.
Note that the observed correlation of the (¯xk,i)k does not cause any
problems in the proof as the submartingale construction does not require inde- pendence. In fact, even the processeszk(n) from the proof of Theorem 6.1 were
not independent due to the common interarrival timesti.
Figure 6.8 shows a numerical illustration of the tightness of the bounds on the waiting time distribution from Theorem 6.8. The illustrated results are for the example of exponentially distributed interarrival and service times with parametersλandµ, respectively.
By combining the above consideration of the compound service time distribution with the results from Section 6.3, one can extend the analysis of random partial mapping to the case of non-renewal input.
Theorem 6.9. (Random Mapping, Non-Renewals, Non-Blocking) Given a FJ queueing system with K parallel non-blocking servers, Markov modulated job interarrivals tj as in Section 6.3, and task service times x¯k,i that are de- scribed by Eq. (6.36). Jobs are randomly mapped to servers according to a uni- form distribution over the set of server combinations with cardinality H. The steady-state waiting and response time distributions are bounded by
P[w≥σ]≤He−θ∗σ , P[r≥σ]≤HE h eθ∗x1,1ie−θ∗σ , whereθ∗ is defined by θ∗:= sup θ >0 (1−pd) +pdEeθx1,1Λ(θ) = 1 . (Recall thatΛ(θ)was defined as a spectral radius of Tθ in Section 6.3).
Proof. The proof follows analogously to the proof of Theorem 6.3 with the dif- ference thatxk,i is replaced by ¯xk,i andK byH, respectively.
Remark 6.10. Random number of serversH: One variation of the system that is considered in Section 6.4.2 is a random mapping of arriving jobs to a random number of servers 1≤H ≤N based on a uniform distribution over the power set 2A
\ {∅} with A = {1, . . . , N}. In this case the steady state waiting and
response times are bounded by
P[w≥σ]≤Ke−θ∗σ ,
P[r≥σ]≤KEheθ∗x1,1ie−θ∗σ ,
whereθ∗ is the solution of (6.39) withp