Upper Bounds on Performance Measures of Heterogeneous Queues

Abstract

In many real-life queueing systems, the servers are often heterogeneous, namely they work at different rates. This paper provides a simple method to compute tight upper bounds on two important performance measures of single-class heterogeneous multi-server Markovian queueing systems, namely the average number in queue and the average waiting time in queue. This method is based on an expansion of the state space that is followed by an approximate reduction of the state space, only considering the most probable states. In most cases tested, we were able to approximate the actual behavior of the system with smaller errors than those obtained from traditional homogeneous multiserver Markovian queues, as shown by GPSS simulations. In addition, we have correlated the quality of the approximation with the degree of heterogeneity of the system, which was evaluated using its Gini index. Finally, we have shown that the bounds are robust and still useful, even considering quite different allocation strategies. A large number of simulation results show the accuracy of the proposed method that is better than that of classical homogeneous multiserver Markovian formulae in many situations.

Scheduling internet of things applications in cloud computing

“…Thus, the lower and upper bounds of the performance metrics, average occupancy, average delay, drop rate, and throughput (n C 1 , δ C 1 , D C 1 , and γ C 1 ) are computed for a heterogeneous multiserver system [37]. The performance metrics of C 1 are only derived by using the same approach with [38] because the performance metrics of C 2 can similarly be derived. Figure 7 shows the state transition diagram of the proposed model.…”

“…State probability equations until mth state can be written as follows by using the state transition diagram [37][38][39] in Fig. 7:…”

“…The values of the service rates in Table 2 represent the only service time of a request in the servers. The values of service rates are selected to present heterogeneous server environment [38]. Creation and preparation times of VMs are ignored because IoT type requests are generally software type requests and do not experience delays due to creation or preparation of VMs.…”

Minimizing the Loss Probability in M/M/2/1 Queueing System with Ordered Entry

“…The most effective measurement of the system is the loss probability, and the loss probability occurs when the two servers are busy and the waiting room has a customer. The steady state probabilities of this system are obtained by the formula: [3] [4] Methods: In this paper, we get the steady state probabilities of this system by formula (1), get the total probability by formula (7), get Laplace-Stieltjes transform by formula (8), get steady state probabilities j π by formula j = 0, 1, 2, 3 by formulas at (IV).…”

“…The probability of lost customers in the queueing system under examination was computed. Furthermore, by using inequality shown that the loss probability was minimum when inter-arrival times fit deterministic distribution [1] [2]. …”