The approximation method of task sojourn time in the fork-join queuing system based on relationship invariants is proposed. The idea is in application of intuitive proportion between the certain queuing systems time characteristics. It is shown that, compared with the known methods, the proposed approximation is more accurate with server utilization higher then 0.7.
To evaluate the efficiency of query processing in systems using distributed and parallel computing technologies, queuing systems like Split-Merge and Fork-Join are used.
The general idea of the Split-Merge and Fork-Join systems functioning is as follows: tasks received in the system are “split” into n sub-tasks, each of which is sent to the channel with numbers 1, 2, ..., n, respectively. Processed subtasks fall into the synchronization buffer, where they wait for the servicing completion of their related subtasks. At the end servicing of the last related subtask, synchronization occurs, i.e. aggregation, after which task leaves the system. It is regarded that synchronization occurs instantly.
The difference between the Split-Merge and Fork-Join systems is shown in Figures 1 and 2 for the example of 3 service channels. In the case of Fork-Join, the freed channel may be occupied by the subtask of the next task (Figure 1). When organizing the Split-Merge service (Figure 2), a block occurs at the time of task entering, and the freed channels are idle, waiting for the last of the subtasks of the current task to be serviced. A rather large number of works has been devoted to the Fork-Join and Split-Merge processes [Alom2014, Bacc1985, Bacc1989, Fior2015, Flat1979, Harr2003, Khab2019, Nels1988, Olv2014, Ryzh1980, Ryzh2015, Qui2015, Var2002, Varma1994, Wright1992]. For the Split-Merge system in [Harr2003], an exact solution was obtained to determine the maximum service time of independent channels with exponential service time and various intensities, as well as approximations for the case of a general distribution. In [Fior2015], the mentioned distribution was obtained for homogeneous and heterogeneous servers, and its representation in matrix-exponential form made it possible to find both the first and the moments of higher orders. It should be noted that this method is characterized by high computational complexity, which is a consequence of the laborious operations of inversion and the Kronecker product of matrices that are included in it. The use of Kronecker algebra is associated with a significant additional memory consumption, as well as many redundant operations with zero operands. In [Ryzh2019], an exact solution was found for an arbitrary distribution of services based on ChebyshevLaguerre numerical integration. The solution has a relatively low complexity and high accuracy. For Fork-Join systems, the exact expression for the average sojourn time with an arbitrary service distribution was obtained only for a system with two channels [Bacc1985, Fior2015]. For the case n > 2 and exponential service, using various methods, approximations of the average sojourn time were obtained in [Nels1988, Var2002, Varma1994]. We give below a brief summary of the results.
In [Flat1979], an exact formula is given for the average sojourn time for n = 2, and the exponential distribution of the service time
ae v1 = çè H 2 r ö
÷ × v1(M ) , 8 ø n where H n = åi=1
1/ i for n = 2 , H 2 = 1.5 , r = l / µ – system utilization, v1(M ) = 1 (µ - l ) – M / M /1 queueing system.
average sojourn time in In [Nels1988], Nelson and Tantawi proposed an approximation for the average sojourn time in a ForkJoin queueing system with n service channels é H v1 » ê n + ë H 2 4 ae 11 èçç1
H ö ù 12 - r
n ÷÷r ú H 2 ø û 8
1 µ - l , n ³ 2.
(2) In [Var2002], Varki and Merchant proposed a formula v1 » 1 é
êH n + µ ë r
ae n 2(1 - r ) èçç åi=1 i - r 1
+ n 1 öù + (1 - 2r )å i=1 i(i - r ) ø÷÷ûú, n ³ 2.
(1) (3) In [Varma1994], Varma proposed a method for approximating the sojourn time based on a combination of high and low system utilization interpolation methods: v1 » [H n + (Vn - H n )r ] 1 µ - l (4) where Vn = åin=1 çèaeç ni øö÷÷(-1)i-1 åim=1 ççèae mi ÷÷øö (mi m-+11)! .
In this paper, we propose a method for finding the average sojourn time for a Fork-Join queueing system with an arbitrary service distribution based on the approximation of relationship invariants.
In [Ryzh2015], to find the time characteristics of multichannel queueing system with priorities, a method was used based on the invariants idea of the relationship between the desired characteristics:
M k / Gk / n » M k / Gk /1× All designations are given in Kendall notation. Calculation methods for the systems indicated on the right are considered known. In particular, iterative methods of Takahashi-Takami or matrix-geometric progression are used to calculate systems [Ryzh1980, Taka1976]. For the non-priority systems M / G / 1 and M / G / n with a non-uniform task flow, the total intensity of the task flow and the weighted average moments of the servicing distribution were used. As an approximation of the service time, it is proposed to use a second-order hyperexponential distribution H 2 applicable for values of the coefficient of variation, both smaller and larger than 1.0. For example, when replacing the gamma distribution with the parameter 1 < a < 2 by H 2 -approximation, one of the “probabilities” will be negative, and the other will exceed 1.0. As computational experiments show [Ryzh2015], these paradoxical intermediate results do not interfere with the successful calculation of queueing system using known numerical methods [Taka1976].
Here we present the scheme proposed in [Ryzh2015] for the implementation of conversion invariants for an n-channel priority system:
a) By multiplying the channel rate by n, we obtain the equivalent single-channel system rate.
b) Calculate the waiting time distribution moments{wi, j } for all types of requests i and the order of the moments j = 1, 3.
c) For the same initial data on the weighted average service times and the total intensity of the incoming flow L as applied to the M / H 2 / 1 system, calculate the number of requests stationary distribution { pi } and the average requests number in the queue: ¥ Q(1) = å (i - 1) pi .
i=1 d) For the M / H 2 / n system, it is similar to carry out the calculation of state probabilities at initial service intensities and obtain the average queue length: ¥ Q(n) = å (i - n) pi .
i=n+1 e) Recalculate the waiting time distribution moments in the system with priorities for all i and j for the multichannel case:
Wi, j = wi, j × Q(n) Q(1) .
f) Using them, according to relation (5), obtain the sojourn time distribution moments of each task type in a multichannel system.
We propose a similar approach for finding the average sojourn time in the Fork-Join queueing system with the number of channels. We rewrite the proportion (5) in the following form
FJ n » FJ 2 × Here FJ n and FJ 2 – Fork-Join queuing systems with n and 2 channels, respectively. To find the average sojourn time in FJ 2 , we use the formula (1). The calculation of the remaining queueing systems from the right-hand side of (6) will be carried out by known methods [Ryzh1980, Taka1976]. To ensure the equality of system utilization values, when calculating M / G / n queuing systems, the initial moments of service time for were multiplied by n 2 .
The following are the results of calculations of the average sojourn time in the queueing system (Figures 3-8) in comparison with the methods proposed in [Nels1988,Var2002,Varma1994] (formulas (2), (3), (4)) and simulation data. The comparison was carried out depending on the system utilization and the number of service channels at three values of the service time coefficient of variation {0.5, 1.0, 1.5}. As can be seen from the figures, the accuracy of approximation by the method of invariants of the ratio is higher than calculated by the formulas (2-4). Moreover, at low ρ values, the method of invariant relations gives underestimated estimates of the average sojourn time, especially when increasing the coefficient of variation of the service time. Since the proposed method (as well as other approximation methods compared here) allows us to estimate only the average sojourn time of task in the system, if higher moments are necessary, the Split-Merge calculation method mentioned above should be used, since the estimated values of the sojourn time estimates in this case are the upper bound stay in Fork-Join queuing system.
The relation invariants method showed well average sojourn time approximation accuracy for Fork-Join queueing system. Calculation results and simulation data comparison showed that the proposed approximation accuracy increases as system utilization increasing. At high the service time coefficient of variation values, the method gives underestimated sojourn time of the task in the system. If it is necessary to find the upper bounds of sojourn time higher moments, it is recommended to use the method proposed in [Khab2019,Ryzh2019]
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