Multiserver Queues with Finite Capacity and Setup Time Tuan Phung-Duc Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo, Japan E-mail:
[email protected]
Abstract. Multiserver queues with setup time have been extensively studied because they have application in modelling of power-saving data centers. Although the infinite buffer models are extensively investigated, less attention has been paid to finite buffer models. This paper considers an M/M/c/K queue with setup time for which we suggest a simple and numerically stable recursion for the stationary distribution of the system state. Numerical experiments show various insights into the performance of the system such as performance-energy tradeoff as well as the effect of the capacity on the blocking probability and the mean queue length. Keywords: multiserver queue, setup time, finite capacity
1
Introduction
The core part of cloud computing is data center where a huge number of servers are available. These servers consume a large amount of energy. Thus, the key issue for the management of these server farms is to minimize the power consumption while keeping acceptable service level for users. It is reported that under the current technology an idle server still consumes about 60% of its peak processing jobs [1]. Thus, the only way to save power is to turn off idle servers. However, off servers need some setup time to be active during which they consume energy but cannot process jobs. Thus, there exists a trade-off between power-saving and performance which could be analyzed by queueing models with setup time. Recently, motivated by applications in data centers, multiserver queues with setup times have been extensively investigated in the literature. In particular, Gandhi et al. [3] extensively analyze multiserver queues with setup times. They obtain some closed form approximations for the ON-OFF policy where any number of servers can be in the setup mode at a time. As is pointed out in Gandhi et al. [3], from an analytical point of view the most challenging model is the ONOFF policy where the number of servers in setup mode is not limited. Gandhi et al. [4, 5] analyze the M/M/c/Setup model with ON-OFF policy using a recursive renewal reward approach. Phung-Duc [11] obtains exact solutions for the same model via generating functions and via matrix analytic methods. Slegers
2
Phung-Duc
et al. [6] propose a heuristic method to decide the timing for the servers to be powered up or down. Although, the infinite model has been investigated [4, 5, 11], results for systems with a large number (several hundreds) of servers are not obtained. This motivates us to develop models for large-scale server farms. Furthermore, less attention has been paid on finite buffer multiserver queue with setup time. It should be noted that the results for the latter could be used for the former by letting the capacity tend to infinity. The main aim of our current paper is to present a simple recursion for the stationary distribution of the M/M/c/K/Setup model which is more realistic for data centers which typically have a finite buffer. The computational complexity of the scheme is significantly reduced in comparison with that of direct methods. As a result, models with several hundreds of servers are easily analyzed. This allows us to explore new insight into the performance of large scale systems that has not been observed in literature. Recently, we become aware of a closely related paper [2], where the authors suggest a recursive scheme for finite buffer model with threshold control. However, the stability of the numerical scheme is not discussed. In contrast to [2], we suggest here a new recursive scheme whose numerical stability is rigorously proved. The rest of this paper is organized as follows. Section 2 presents the model in details while Section 3 is devoted to derivation of a recursion for the joint stationary distribution. Section 4 presents some numerical examples showing insights into the performance of the system. Concluding remarks are presented in Section 5.
2
Model
We consider a queueing system with c servers and a capacity of K, i.e., the maximum of K customers can be accommodated in the system. Jobs arrive at the system according to a Poisson process with rate λ. In this system, a server is turned off immediately if it has no job to do. Upon arrival of a job, an OFF server is turned on if any and the job is placed in the buffer. However, a server needs some setup time to be active so as to serve waiting jobs. We assume that the setup time follows an exponential distribution with mean 1/α. Let j denotes the number of customers in the system and i denotes the number of active servers. The number of servers in setup process is min(j − i, c − i). Under these assumptions, the number of active servers is smaller than or equal to the number of jobs in the system. Therefore, in this model a server is in either BUSY or OFF or SETUP. We assume that the service time of jobs follows an exponential distribution with mean 1/µ. We assume that waiting jobs are served according to a first-come-first-served (FCFS) manner. We call this model an M/M/c/K/Setup queue. The exponential assumptions for the inter-arrival, setup time and service time allow to construct a Markov chain whose stationary distribution is recursively obtainable. It should be noted that we can easily construct a Markov chain for a more general model with MAP arrival and phase-type service and setup time
Multiserver Queues with Finite Capacity and Setup Time
3
distributions. However, the state space of the resulted Markov chain explodes and thus the analysis is complex.
3
Analysis
In this section, we present a recursive scheme to calculate the joint stationary distribution. Let C(t) and N (t) denote the number of active servers and the number of customers in the system, respectively. It is easy to see that {X(t) = (C(t), N (t)); t ≥ 0} forms a Markov chain on the state space: S = {(i, j); 0 ≤ i ≤ c, j = i, i + 1, . . . , K − 1, K}. See Figure 1 for transition among states for the case c = 2 and K = 5.
Fig. 1. Transition among states (c = 2, K = 5).
Let πi,j = limt→∞ P(C(t) = i, N (t) = j) ((i, j) ∈ S) denote the joint stationary distribution of {X(t)}. In this section, we derive a recursion for calculating the joint stationary distribution πi,j ((i, j) ∈ S). The balance equations for states with i = 0 read as follows. λπ0,0 = µπ1,1 , (λ + min(j, c)α)π0,j = λπ0,j−1 , cαπ0,K = λπ0,K−1 .
j = 1, 2, . . . , K − 1,
4
Phung-Duc (0)
(0)
= λ/(λ + min(j, c)α) (j = 1, 2, . . . , K − 1)
leading to π0,j = bj π0,j−1 where bj (0) bK
and = λ/(cα). Furthermore, it should be noted that π1,1 is calculated using the local balance equation in and out the set {(0, j); j = 0, 1, . . . , K} as follows. µπ1,1 =
K ∑
min(j, c)απ0,j .
j=1
Remark 1. We have expressed π0,j (j = 1, 2, . . . , K) and π1,1 in terms of π0,0 . Next, we consider the case i = 1. Lemma 1 We have (1)
(1)
j = 2, 3, . . . , K − 1, K,
π1,j = aj + bj π1,j−1 , where (1)
(1)
aj
(1)
bj
= =
µaj+1 + min(j, c)απ0,j (1)
µ + λ + min(j − 1, c − 1)α − µbj+1 λ (1)
µ + λ + min(j − 1, c − 1)α − µbj+1
,
(1)
,
(2)
for j = K − 1, K − 2, . . . , 2 and (1)
aK =
cαπ0,K , µ + (c − 1)α
(1)
bK =
λ . µ + (c − 1)α
Proof. We prove using mathematical induction. Balance equations are given as follows. (λ + µ + min(j − 1, c − 1)α)π1,j = λπ1,j−1 + µπ1,j+1 + min(j, c)απ0,j , (µ + min(K − 1, c − 1)α)π1,K
2 ≤ j ≤ K − 1, = λπ1,K−1 + cαπ0,K .
(3) (4)
It follows from (4) that (1)
(1)
π1,K = aK + bK π1,K−1 , leading to the fact that Lemma 1 is true for j = K. Assuming that Lemma 1 (1) (1) is true for j + 1, i.e., π1,j+1 = aj+1 + bj+1 π1,j . It then follows from (3) that (1)
(1)
Lemma 1 is also true for j, i.e., π1,j = aj + bj π1,j−1 . Theorem 2 We have the following bound. (1)
aj
≥ 0,
for j = 2, 3, . . . , K − 1, K.
(1)
0 ≤ bj
≤
λ , µ + min(j − 1, c − 1)α
Multiserver Queues with Finite Capacity and Setup Time
5
Proof. We use mathematical induction. It is easy to see that the theorem is true for j = K. Assuming that the theorem is true for j + 1, i.e., (1)
(1)
aj+1 ≥ 0,
0 ≤ bj+1 ≤
λ , µ + min(j, c − 1)α
j = 1, 2, . . . , K − 1.
(1)
Thus, we have µbj+1 < λ. From this inequality, (1) and (2), we obtain (1)
bj (1)
and aj
≤
λ , µ + min(j − 1, c − 1)α
≥ 0.
It should be noted that π2,2 can be calculated using the local balance between the flows in and out the set of states {(i, j); i = 0, 1, j = i, i + 1, . . . , K} as follows. K ∑ 2µπ2,2 = min(j − 1, c − 1)απ1,j . j=2
Remark 2. We have expressed π1,j (j = 1, 2 . . . , K) and π2,2 in terms of π0,0 . We consider the general case where 2 ≤ i ≤ c − 1. Similar to the case i = 1, we can prove the following result by mathematical induction. Lemma 3 We have (i)
(i)
πi,j = aj + bj πi,j−1 ,
j = i + 1, i + 2, . . . , K − 1, K,
where (i)
(i)
aj =
iµaj+1 + min(c − i + 1, j − i + 1)απi−1,j (i)
λ + min(c − i, j − i)α + iµ − iµbj+1 λ
(i)
bj = and (i)
aK =
(i)
λ + min(c − i, j − i)α + iµ − iµbj+1
(c − i + 1)απi−1,K , (c − i)α + iµ
(i)
bK =
,
,
λ . (c − i)α + iµ
Proof. The balance equation for state (i, K) is given as follows. ((c − i)α + iµ)πi,K = λπi,K−1 + (c − i + 1)απi−1,K , leading to the fact that Lemma 3 is true for j = K. Assuming that (i)
(i)
πi,j+1 = aj+1 + bj+1 πi,j ,
j = i + 1, i + 2, . . . , K − 1.
(5) (6)
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Phung-Duc
It then follows from (λ + min(c − i, j − i)α + iµ)πi,j = λπi,j−1 + iµπi,j+1 + min(c − i + 1, j − i + 1)απi−1,j , j = K − 1, K − 2, . . . , i + 1, that (i)
(i)
πi,j = aj + bj πi,j−1 . Theorem 4 We have the following bound. λ , iµ + min(j − i, c − i)α
(i)
(i)
0 < bj
0,
for j = i + 1, i + 2, . . . , K − 1, i = 1, 2, . . . , c − 1. Proof. We also prove using mathematical induction. It is clear that Theorem 4 is true for j = K. Assuming that Theorem 4 is true for j + 1, i.e., (i)
(i)
aj+1 > 0,
0 < bj+1
0,
(c)
0 < bj
0,
(c)
0 < bj+1
