with staggered setup
Introduction
We consider an system, where there is a setup cost required to turn on a server which is currently turned off. Setup costs are common in manufacturing systems [1], where there is often a “warmup” time needed to get a machine running, or a “transport time” needed when calling a staff member into work. Setup costs are also common in data centers [4], where there is a “boot up” time needed to turn on an off server.
In data centers, idle servers are often turned off to save power. However, turning on an off server requires a setup cost. Setup costs are wasteful in two ways: (i) they impose a setup time, which is the time it takes the server to turn on, thereby increasing overall mean response time, (ii) they waste power, since peak power is consumed during the setup time, although no work is being done. To save on power, the number of servers that can be in setup at any time is often purposely limited. In the staggered setup model [6], [1], at most one server can be in setup at any time.
We consider an system where jobs arrive according to a Poisson process with rate and are served at rate , where denotes the job size. For stability, we assume that .
Each of the servers is in one of three states: off, on (being used to serve a job), or setup (undergoing the setup cost). In the model that we consider, we allow at most one server to be in setup at any given time. When servers are not in use, they are immediately switched to the off state. When a new job arrives, if there is already a server in the setup state, then the job simply joins the queue, otherwise the job picks an off server (assuming there is one) and switches it into the setup state. We use to denote the setup times, with . Unless stated otherwise, we assume that the setup times are exponentially distributed, with rate . When a job completes service at a server, , the job at the head of the queue is moved to server , without the need for setup, since server is already on. Note that even if the job at the head of the queue was already waiting on another server in setup mode, the job at the head of the queue is still directed to server . At this point, if there is another job in the queue, then server continues to be in setup for this job. If no such job exists in the queue, then server is turned off.
While there has been a lot of work on single-server systems with setup costs, there has been very little work on multi-server systems with setup costs. For the single-server, [5], in 1986, showed that the distribution of response time for an system with setup times, referred to as , has the following decomposition property: where denotes the setup time, and is exponentially distributed. Note that in the case of a single-server system, is the same as . For multi-server queues, only recently in 2010, [4] showed that the distribution of response time for an with exponential setup time, , has the following decomposition property: However, no results exist for the system.
In this paper, we present results suggesting that the decomposition property in Eq. (2) provides a very good approximation for the system, with exponential setup times. That is, the distribution of response time for an can be well approximated as: In Section 2, we prove that the decomposition property, as in Eq. (3) above, holds exactly for the , where the job size distribution, , is a degenerate exponential. Then, in Section 3, we present matrix analytic results suggesting that the decomposition property provides a very good approximation (nearly exact) for the and the , where and are the hyper-exponential and the hypo-exponential job size distributions respectively. In Section 4, we present simulation results suggesting that the decomposition property provides a very good approximation for various job size distributions including Deterministic, Uniform, Weibull, Bounded Exponential, and Bounded Pareto. Finally, we conclude in Section 5 with a discussion of the system, and the system with non-exponential setup times and non-staggered setup.
Section snippets
The distribution is the degenerate exponential distribution, whereby with probability , the job size is zero, and with probability , the job size is exponentially distributed with mean . Thus, the overall mean job size is . The squared coefficient of variation for the is . The is an important distribution in queuing theory because its value spans the entire range from 1 to , allowing it to represent a variety of job size distributions. We now prove the
and
The is the hyper-exponential job size distribution, whereby with probability , the job size is of type I (exponential with mean ), and with probability , the job size is of type II (exponential with mean ). The distribution is far broader than the distribution.
The can be analyzed numerically, via matrix analytic methods. While the Markov chain is complex (and is thus omitted due to lack of space), it is tractable via matrix analytic methods due to its regular
In this section, we demonstrate via simulations that the decomposition property provides a very good approximation for the , when follows distributions other than the hyper-exponential and hypo-exponential. Each simulation run consists of 107 arrivals, and we average our results over multiple runs for each job size distribution.
Fig. 1 shows a subset of our simulation results for the first four moments of response time for an system (additional results can be found in [3]). In
Conclusion
In this paper, using analysis, matrix analytic methods, and simulations, we show that the response time distribution for an ( with exponential staggered setup) can be well approximated using the sum of the setup time distribution and the response time distribution of an without setup, as in Eq. (3). For the case when , we prove that this approximation is exact.
The fact that the setup time is exponentially distributed is important. We can prove [3] that the
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