Nurşen Aydın, İbrahim Muter and İlker Birbil

This repository accompanies our paper "Multi-objective temporal bin packing problem: An application in cloud computing" published in Computers & Operations Research, Volume 121, 2020, 104959, ISSN 0305-0548.

Improving energy efficiency and lowering operational costs are the main challenges faced in systems with multiple servers. One prevalent objective in such systems is to minimize the number of servers required to process a given set of tasks under server capacity constraints. This objective leads to the well-known bin packing problem. In this study, we consider a generalization of this problem with a time dimension, where the tasks are to be performed with predefined start and end times. This new dimension brings about new performance considerations, one of which is the uninterrupted utilization of servers. This study is motivated by the problem of energy efficient assignment of virtual machines to physical servers in a cloud computing service. We address the virtual machine placement problem and present a binary integer programming model to develop different assignment policies. By analyzing the structural properties of the problem, we propose an efficient heuristic method based on solving smaller versions of the original problem iteratively. Moreover, we design a column generation algorithm that yields a lower bound on the objective value, which can be utilized to evaluate the performance of the heuristic algorithm. Our numerical study indicates that the proposed heuristic is capable of solving large-scale instances in a short time with small optimality gaps.

Keywords: Bin packing; Cloud computing; Heuristics; Exact methods; Column generation

We provide 240 problem instances. For all of the test problems, the names of the input files are of the form cap100_n#_t#_dici_no.txt, where cap100, n, t, di, ci and no are defined as follows:

• cap100: In all instances, we assume that the servers are identical and the capacity is set to Cap = 100.

• n# : the number of virtual machine requests (e.g. n50, 50 VM requests).

• t#: the length of the VM arrival period (e.g. t50, VMs arrive between time 1 and time 50).

• di: the length of requested time interval which can be considered as the duration of a VM request. We define two sets for VM duration, denoted by dS and dL, which are randomly generated in intervals [10, 30] and [20, 60], respectively. In the input files, dS is denoted by Sh, dL is denoted by Lon.

• ci: the requested capacity for a VM request. We define two parameters, namely cmin (minimum capacity demand) and cmax (maximum requested capacity). We generate integer capacity request for a VM randomly between cmin and cmax. In our experiments, we set cmin = 25 and test the models for varying capacity requirements by changing cmax ∈ {50,75} In the input files, cmax = 50 is denoted by Sm, cmax = 75 is denoted by Lr.

• no: number of the instance file. For each combination of these parameters, 5 instances were generated.

For instance, the input file "cap100_n100_t120_LonLr_1.txt" has 100 VM requests with the length of the arrival period of 120. The durations of VM requests (dL) are randomly generated in intervals [20, 60] and cmax is set to 75.

Content of the input file:

• The first line in the input files provides the number of VM requests and the capacity of one server. For instance, in the input file "cap100_n100_t120_LonLr_1.txt" 100 100 0 0 --> 100 is the number of VM requests, 100 is the capacity of one server

• From the second line to the end of the file, it provides the data for each arrived VM request. The first column corresponds to the VM number (starts with 0). The second column is the starting time of the request. The third column is the completion time of the VM request. The last column corresponds to the capacity request.

  For instance, in the input file "cap100_n100_t120_LonLr_1.txt", the second line is given as: 0 2 51 60 --> VM no=0, the starting time=2, the completion time=51 and the capacity request=60.