Robust Queueing Theory - CiteSeerX

Submitted to Operations Research manuscript (Please, provide the mansucript number!)

Robust Queueing Theory Chaithanya Bandi*, Dimitris Bertsimas† , Nataly Youssef‡ We propose an alternative approach for studying queueing systems by employing robust optimization as opposed to stochastic analysis. While traditional stochastic queueing theory relies on Kolmogorov’s axioms of probability and models arrivals and services as renewal processes, we use the limit laws of probability as the axioms of our methodology and model the queueing systems primitives by uncertainty sets. In this framework, we obtain closed form expressions for the steady-state waiting times in multi-server queues with heavy-tailed arrival and service processes. These expressions are not available under traditional stochastic queueing theory for heavy-tailed processes, while they lead to the same qualitative insights for independent and identically distributed arrival and service times. We also develop an exact calculus for analyzing a network of queues with multiple servers based on the following key principle: a) the departure from a queue, b) the superposition, and c) the thinning of arrival processes have the same uncertainty set representation as the original arrival processes. We show that our approach, which we call the Robust Queueing Network Analyzer (RQNA) a) yields results with error percentages in single digits (for all experiments we performed) relative to simulation, b) performs significantly better than the Queueing Network Analyzer (QNA) proposed in Whitt (1983), and c) is to a large extent insensitive to the number of servers per queue, the network size, degree of feedback, traffic intensity, and somewhat sensitive to the degree of diversity of external arrival distributions in the network. Key words : Queueing Theory, Robust Optimization, Heavy Tails, Stochastic Networks

1. Introduction The origin of queueing theory dates back to the beginning of the 20th century, when Erlang (1909) published his fundamental paper on congestion in telephone traffic. In addition to formulating and solving several practical problems arising in telephony, Erlang laid the foundations for queueing theory in terms of the nature of assumptions and techniques of analysis that are being used to this ∗

Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139, [email protected]

†

Boeing Professor of Operations Research, Co-director, Operations Research Center, Massachusetts Institute of

Technology, Cambridge,Massachusetts 02139, [email protected] ‡

Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139, [email protected] 1

2

Author: Robust Queueing Theory Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!)

day. In the second part of the 20th century, a very substantial literature of queueing theory was developed modeling queueing primitives as renewal processes. From the time of Erlang, the Poisson process has played a very significant role in modeling the arrival process of a queue. When combined with exponentially distributed service times, the resulting M/M/m queue with m servers is tractable to analyze in steady-sate. While exponentiality leads to a tractable theory, assuming general distributions, on the other hand, yields considerable difficulty with respect to performing a near-exact analysis of the system. The GI/GI/m queue with independent and generally distributed arrivals and services is, by and large, intractable. Currently, there does not exist a method that is capable of producing accurate numerical answers, let alone closed form expressions, for arbitrary distributions. The most general method, due to Pollaczek (1957), analyzes the performance of the GI/GI/m queue by formulating a multi-dimensional problem in the complex plane. Gall (1998) portrays the exceptional difficulty of explicitly characterizing the equations for the GI/GI/m queue given that their “partial solution can only be derived after long and complex calculations involving multiple contour integrals in a multi-dimensional complex plane”. When arrival and service distributions have rational Laplace transforms of order p (for example Coxian distributions with p phases), the GI/GI/m problem becomes intractable for higher order p values. Bertsimas (1990) reports numerical results for queues with up to 100 servers and p = 2 by finding all h = m+p−1 complex roots to distinct polynomial equations and solving m a linear system of dimension h. The system’s dimension, however, increases to 4.5 million when p = 5, hence illustrating the complexity of the problem under these assumptions. The situation becomes even more challenging if one considers analyzing the performance of queueing networks. A key result that allows generalizations to networks of queues is Burke’s theorem (Burke (1956)) which states that the departure process from an M/M/m queue is Poisson. This property allows one to analyze queueing networks and leads to product form solutions as in Jackson (1957). However, when the queueing system is not M/M/m, the departure process is no longer a renewal process, i.e., the interdeparture times are dependent. With the departure process lacking the renewal property, the state-of-the-art theory provides no means to determine


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performance measures exactly, even for a simple network with queues in tandem. The two avenues in such cases are simulation and approximation. Simulation can take a considerable amount of time in order for the results to be statistically significant. In addition, simulation models are often complex, which makes it difficult to isolate and understand key qualitative insights. On the other hand, approximation methods can potentially lead to results that are not very close to the true answers. Given these challenges, it is fair to say that the key problem of performance analysis of queueing networks has remained open under the probabilistic framework. In his opening lecture at the conference entitled “100 Years of Queueing–The Erlang Centennial”, Kingman (2009), one of the pioneers of queueing theory in the 20th century, writes, “If a queue has an arrival process which cannot be well modeled by a Poisson process or one of its near relatives, it is likely to be difficult to fit any simple model, still less to analyze it effectively. So why do we insist on regarding the arrival times as random variables, quantities about which we can make sensible probabilistic statements? Would it not be better to accept that the arrivals form an irregular sequence, and carry out our calculations without positing a joint probability distribution over which that sequence can be averaged?”. In practice, probability distributions are not inherent to the queueing system; they represent a modeling choice of the modeler that attempts to approximate the actual underlying behavior of the arrival and service processes. We propose an alternative framework to model queueing systems based on optimization theory. The motivation behind our idea stems from the rich development of optimization as a scientific field during the second part of the 20th century. From its early years (Dantzig (1949)), modern optimization has had the objective to solve multi-dimensional problems efficiently from a practical point of view. Today, many commercial codes are available which can solve truly large scale structured (linear, mixed integer and quadratic) optimization problems. In particular, Robust Optimization (RO), arguably one of the fastest growing areas in optimization in the last decade, provides, in our opinion, a natural modeling framework for stochastic systems. For a review of robust optimization, we refer the reader to Ben-Tal et al. (2009), and Bertsimas et al. (2011a). The key idea of our


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approach is to make the limit laws of probability theory the primitive assumptions and formulate the problems arising in queueing systems as robust optimization problems. An initial effort along these lines includes the work by Bertsimas et al. (2011b) where probabilistic guarantees on the length of a busy period and the waiting time are provided through robust optimization. Herein, we build upon this work and present a new approach for modeling the primitives of queueing systems by uncertainty sets. This framework allows us to derive exact performance analysis of the underlying stochastic system. The present paper is part of a broader investigation to analyze stochastic systems using robust optimization: performance analysis of multi-class queueing networks (Bandi et al. (2012a)), performance analysis of queueing systems in the transient domain (Bandi et al. (2012b)), network information theory (Bandi and Bertsimas (2012b)), multi-item, multi-bidder auctions (Bandi and Bertsimas (2011)) and pricing of multi-dimensional derivative securities (Bandi et al. (2011)). For a survey of the overall approach, see Bandi and Bertsimas (2012a). Our robust optimization approach to queueing theory bears philosophical similarity with the deterministic network calculus approach which was pioneered by Cruz (1991a,b) (see also Gallager and Parekh (1993, 1994), El-Taha and Stidham (1999), C.S.Chang (2001), Boudec and Thiran (2001) ). Both methods (a) take a non-probabilistic approach by placing deterministic constraints on the traffic flow, (b) derive bounds on key queueing performance measures via a worst-case paradigm, and (c) allow a tractable performance analysis of queues. In a similar context, Borodin et al. (2001), Gamarnik (2003, 2000), Goel (1999) undertakes a deterministic worst-case approach via adversarial queueing theory to analyze the stability of multi-class queueing networks. While philosophically similar, the aspirations we have through this work does not encompass stability analysis, and therefore we believe there is no overlap between the two approaches. Beyond their deterministic and worst-case paradigms, significant differences can be noted when comparing our framework to the network calculus approach. (a) Different Underlying Assumptions: While both methods postulate deterministic constraints over the arrival process, the assumptions are different in nature. The deterministic


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network calculus bounds the number of external arrivals nt up to time t by nt ≤ λ · t + B, where λ denotes the traffic rate and B is a constant accounting for burstiness. In contrast, our assumption on the arrival process yields different bounds on the number of arrivals nt . In fact, denoting the arrival time of the nth t job by t, i.e., tion 1(a) with tail coefficient αa = 2, we obtain nt −

√

Pnt

i=1 Ti

= t, and applying Assump-

nt λΓa ≤ λt ≤ nt +

√

nt λΓa , where Γa

represents the effect of variability. Writing δ 2 = nt yields δ 2 − λΓa δ ≤ λt ≤ δ 2 + λΓa δ. This p 1 3 implies that δ ≥ −λΓa + λ2 Γ2a + 4λt /2, leading to nt ≥ λt − t 2 λ 2 Γa . Similarly, we obtain 1

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nt ≤ λt + t 2 λ 2 Γa , which results in the following bounds on the number of arrivals by time t |nt − λ · t| ≤ Γa λ3/2 t1/2 .

(1)

Note that the way we handle variability is different from the deterministic network calculus, and is motivated and indeed consistent with the limit laws of probability (see Section 2.2). (b) Tighter Bounds: It is widely believed that the network calculus approach can provide overly conservative bounds for single-server queues. Ciucu and Hohlfeld (2010) attribute this conservatism to the fact that the statistical properties of the arrivals are not accounted for in the assumptions of the deterministic network calculus. To remedy this, stochastic network calculus was developed (see Jiang and Liu (2008), Jiang (2012), Ciucu et al. (2005), Burchard et al. (2011) for an overview), positing probabilistic assumptions over the queueing primitives. This allowed reasonably accurate bounds compared with M/M/1 and M/D/1 queueing systems (see Ciucu (2007)). Our approach, however, provides a bound on the waiting time for single-server queues that is qualitatively similar to its probabilistic counterpart (see Section 3.3). Our computations further show that, by constraining nature via bounding the variability allowed in our uncertainty sets, we obtain results within 8% of simulated stochastic queues and queueing networks (see Section 7). (c) Generalizability: Our approach generalizes the analysis to more complex queueing systems such as multi-server queues (see Section 3.2) and queueing networks with feedback (see Section 5). However, “for GI/GI/m, (m > 1), stochastic network calculus based analysis remains plain

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blank ” and “feedback analysis is perhaps the most critical open challenge for stochastic network calculus”, as remarked by Jiang (2012). Furthermore, while the stochastic network calculus has recently addressed heavy tails in a single-server setting (see Burchard et al. (2012)), our framework is capable of providing closed-form and tight bounds on the waiting time, while maintaining deterministic assumptions. Specifically, our contributions and structure of the paper are as follows: (a) In Section 2, we introduce the notion of a robust queue as an alternative to the traditional queueing model and propose to replace the renewal process primitives with uncertainty sets that the arrival and service processes satisfy. (b) In Section 3, we analyze the steady-state behavior of single and multi-server robust queues and obtain closed form expressions for the waiting times, which carry the same qualitative insights as traditional queueing theory and extend to include heavy-tailed arrivals and services. (c) Section 4 presents an analog of Burke’s theorem, where the uncertainty set characterizing the departure process is shown to be the same as the uncertainty set characterizing the arrival process. This is a remarkable property of our uncertainty set model which holds in considerable more generality than the M/M/m queue. (d) Armed with the analog of Burke’s theorem, in Section 5, we develop a calculus describing the effect of the three operations characterizing queueing networks on the arrival uncertainty set: arrival superposition, process thinning which models probabilistic routing, and passing through a queue. This allows us to exactly characterize the arrival process of any queue that operates in a queueing network. (e) In Section 6, we analyze queueing networks with asymmetric heavy-tailed arrival and service processes. In particular, we provide an extension of our results in Sections 3-5 to accommodate the case where arrival and service times possess different tail behaviors. (f ) In Section 7, we report computational results for multi-server queueing networks, which suggest that the proposed approach can be adapted to be within 4-6% from simulation. We also report on the sensitivity of the results as a function of the number of servers per queue, the


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network size, degree of feedback, traffic intensity, and the degree of diversity of external arrival distributions in the network.

2. The New Primitives We introduce the notion of a robust queue where we model the arrival and service processes by uncertainty sets instead of assigning probability distributions.

2.1. The Robust Queue Model We denote the interarrival time between the (i − 1)st and ith jobs by Ti and the service time of job i by Xi . We propose the following uncertainty sets on the interarrival and service processes. Assumption 1. We make the following assumptions for the interarrival and service times (a) The interarrival times belong to the uncertainty set n   X (n − k)       Ti −     λ i=k+1 a ≤ Γ , ∀ k ≤ n − 1 , U = (T1 , T2 , . . . , Tn ) a   (n − k)1/αa         where 1/λ is the expected interarrival time, Γa is a parameter that captures variability information and 1 < αa ≤ 2 models possibly heavy-tailed probability distributions. (b) The service times for an m-server belong to the uncertainty set   v X (v − k)       Xim+r −     µ i=k+1 s ≤ Γ , ∀ k ≤ v − 1 Um = (Xm+r , X2m+r , . . . , Xvm+r ) , s   (v − k)1/αs         where 0 ≤ r < m, 1/µ is the expected service time, Γs is a parameter that captures variability information and 1 < αs ≤ 2 models possibly heavy-tailed probability distributions. For the case of a single server queue, that is, when m = 1, the uncertainty set is given by   n X (n − k)       Xi −     µ i=k+1 s ≤ Γ , ∀ k ≤ n − 1 . U = (X1 , X2 , . . . , Xn ) s   (n − k)1/αs        


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Note that we assume that both the interarrival and service times have bounded support as seen by s letting k = n − 1 in the sets U a , Um , and U s . The key idea in the construction of our uncertainty

sets is to make the primitive assumptions follow from the major conclusions of probability theory, namely its asymptotic laws as opposed to the probability axioms. In the next section, we propose constructing the uncertainty sets based on the central limit theorem (CLT) and the stable limit laws.

2.2. Construction of Uncertainty Sets It is possible to motivate the construction of these uncertainty sets through probabilistic weak convergence theorems. These theorems express the distribution of the sum of many independent and identically distributed random variables as converging to one of a small set of stable distributions. (a) Construction Based on the Central Limit Theorem: Suppose that the interarrival and service times are independent and identically distributed with finite standard deviation σa and σs , respectively. By the central limit theorem, as n → ∞, the random variables n X

n X

n−k Ti − λ i=k+1 σa (n − k)1/2

and

Xi −

i=k+1

n−k µ

σs (n − k)1/2

are asymptotically standard normal. We know that a standard normal Z satisfies P(|Z | ≤ 2) ≈ 0.95, P(|Z | ≤ 3) ≈ 0.99. We therefore assume that the quantities Ti and Xi take values such that n X n − k Ti − ≤ Γa (n − k)1/2 λ i=k+1

and

n X n − k Xi − ≤ Γs (n − k)1/2 µ i=k+1

with Γa and Γs are variability parameters that can be adapted to ensure a good empirical fit (see Section 7 for computational results). Note that the tail coefficient inherent to the normal distribution is α = 2. (b) Construction Based on the Stable Limit Laws: Suppose now that the interarrival and service times are independent with undefined variance. The stable limit laws allow us to construct uncertainty sets for heavy-tailed distributions.


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Theorem 1. (Nolan (1997)) Let Y1 , Y2 , . . . be a sequence of i.i.d. random variables, with mean µ and undefined variance. If Yi ∼ Y , where Y is a stable distribution with parameter α ∈ (1, 2] then n X

Yi − nµ

i=1

n1/α

∼ Y.

Note that the n1/α scaling in Theorem 1 motivates the scaling we used in Assumption 1. Intuitively, by using a value of αa , αs < 2, we allow the interarrival and service times to take larger values when compared to αa , αs = 2, thus, allowing us to model the heavy-tailed nature.

We note that while the uncertainty sets are motivated by i.i.d. assumptions on the underlying random variables, (T1 , T2 , . . . , Tn ) ∈ U a does not necessarily imply that (T1 , T2 , . . . , Tn ) are independent. In summary, the key data primitives characterizing (a) the arrival process in the queue are (λ, Γa , αa ); (b) the service process in the queue are (µ, Γs , αs ). We first assume that arrival and service processes have symmetric tail behavior, i.e., αa = αs = α in Sections 3-5, and then provide the generalized results for the asymmetric case in Section 6.

3. The Multi-Server Robust Queue In this section, we analyze the robust queue model with a first-come first-served scheduling policy and a traffic intensity ρ = λ/(mµ) < 1, where m denotes the number of servers in the queue. We denote by Wn the waiting time of the nth job in this system. Kingman (1970) provides insightful bounds on the expected waiting time in steady state for the GI/GI/1 queue λ σa2 + σs2 · , 2 1−ρ

(2)

λ σa2 + σs2 /m + (1/m − 1/m2 )/µ2 · . 2 1−ρ

(3)

E[Wn ] ≤ and for the GI/GI/m queue E[Wn ] ≤


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While E[Wn ] seeks the expected waiting time when nature obeys the axioms of probability, we seek the highest waiting time when nature is constrained to obey the limit laws of probability. Assuming s interarrival times {Ti }i≥1 ∈ U a and service times {Xi }i≥1 ∈ Um , we define the highest waiting time

cn = W

max

s T∈U a ,X∈Um

Wn .

(4)

and the steady-state worst-case waiting time as

c = lim W cn . W

(5)

n→∞

c for robust queues with single and multiple servers for heavy-tailed arrival and We characterize W service processes.

3.1. Waiting Time in a Single-Server Robust Queue c. We consider a robust queue with a single server and provide a closed form expression for W Theorem 2. If {Ti }i≥1 ∈ U a , {Xi }i≥1 ∈ U s , αa = αs = α and ρ < 1 , then α/(α−1)

1/(1−α) (Γa + Γs ) c ≤ α−1 · λ W α/(α−1) α (1 − ρ)1/(α−1)

.

(6)

The waiting time of the nth job can be expressed recursively in terms of the

Proof of Theorem 2.

interarrival and service times using the Lindley recursion (Lindley (1952))

Wn = max (Wn−1 + Xn−1 − Tn , 0) = max

1≤j≤n−1

n−1 X

X` −

`=j

n X

! T` , 0 .

(7)

`=j+1

cn can be written as Thus, W cn = W

max

max

X∈U s ,T∈U a 1≤j≤n−1

n−1 X `=j

X` −

n X `=j+1

! T` , 0 = max

max

1≤j≤n−1 X∈U s ,T∈U a

n−1 X

X` −

`=j

n X

! T` , 0 .

(8)

`=j+1

From Assumption 1, the sums of the service times and interarrival times are bounded by n−1 X

n X n−j n−j 1/α X` ≤ + Γs (n − j) , T` ≥ − Γa (n − j)1/α . µ λ `=j+1 `=j

(9)


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Combining Eqs. (8) and (9), we obtain an one-dimensional concave maximization problem (since 1 < α ≤ 2) 1−ρ 1/α (Γa + Γs ) (n − j) − (n − j) . 1≤j≤n−1 λ

cn ≤ max W

(10)

Making the transformation x = n − j, Eq. (10) becomes cn ≤ max β · x1/α − γ · x, W

(11)

1≤x≤n−1

with β = Γa + Γs and γ = (1 − ρ)/λ > 0, given ρ < 1. Hence, c ≤ max β · x1/α − γ · x ≤ max β · x1/α − γ · x W x≥1

x≥0

(12)

The maximizer in Eq. (12) is given by

∗

x =

β αγ

α/(α−1)

=

λ(Γa + Γs ) α(1 − ρ)

α/(α−1) .

Substituting β and γ by their respective expressions in Eq. (12) yields Eq. (6).

(13)

Remark on Theorem 2 - Tightness of the bound We note that the upper bound on the waiting time in Theorem 2 is in fact tight. Let us first consider the case where the maximizer x∗ in Eq. (13) is an integer, and let j ∗ = n − x∗ . The following sequence of interarrival times Ti =

  1/λ 

1/λ − Γa /(x∗ )1−1/α

i = 1, . . . , j ∗ , (14) i = j ∗ + 1, . . . , n,

and service times Xi =

  1/µ 

1/µ + Γs /(x∗ )1−1/α

i = 1, . . . , j ∗ − 1, (15) i = j ∗ , . . . , n − 1.

lie in the uncertainty sets in Assumption 1. This can be seen via the partial sums given by  n j = 1, . . . , j ∗ − 1,  (n − j)/λ − Γa (x∗ )1/α X Ti =  i=j+1 (n − j)/λ − Γa (n − j)/(x∗ )1−1/α j = j ∗ , . . . , n − 1, and n−1 X i=j

Xi =

  (n − j)/µ + Γs (x∗ )1/α 

(n − j)/µ + Γs (n − j)/(x∗ )1−1/α

j = 1, . . . , j ∗ − 1, j = j ∗ , . . . , n − 1,


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and noting that (n − j)/(x∗ )1−1/α ≤ (n − j)1/α for all j = j ∗ , . . . , n − 1. We then obtain  n−1 n j = 1, . . . , j ∗ − 1,  (Γa + Γs )(x∗ )1/α − (1 − ρ)(n − j)/λ X X Xi − Ti =  i=j i=j+1 (Γa + Γs )(n − j)/(x∗ )1−1/α − (1 − ρ)(n − j)/λ j = j ∗ , . . . , n − 1. Note that, since (Γa + Γs )(x∗ )1/α is a constant with respect to j, max

1≤j≤j ∗ −1

∗ 1/α

(Γa + Γs )(x )

− (1 − ρ)(n − j)/λ

< (Γa + Γs )(x∗ )1/α − (1 − ρ)(x∗ )/λ α/(α−1)

α − 1 λ1/(1−α) (Γa + Γs ) = α/(α−1) · α (1 − ρ)1/(α−1)

.

The first strict inequality is due to the fact that for j ≤ j ∗ − 1 = n − x∗ − 1, we have x∗ < n − j. Moreover, for j ≥ j ∗ , we have n − j ≤ n − j ∗ , and given that x∗ is the maximizer of Eq. (4), we have (Γa + Γs )(n − j)/(x∗ )1−1/α − (1 − ρ)(n − j)/λ ≤ (Γa + Γs )(n − j)1/α − (1 − ρ)(n − j)/λ ≤ (Γa + Γs )(x∗ )1/α − (1 − ρ)(x∗ )/λ.

(16)

Therefore, we have shown that when the maximizer x∗ is an integer, then there exists a sequence of interarrival and service times that exactly achieves a waiting time equal to the bound in Eq. (6). We next show that when the maximizer x∗ is fractional, then the actual highest waiting time is within a fraction O (1 − ρ)α/(α−1) to the bound in Eq. (6). Note that when x∗ is fractional, due to the concave nature of the function in Eq. (12), the optimal value j ∗ of Eq. (10) is either n − dx∗ e or n − bx∗ c. Denoting the function in Eq. (12) by f (x) = βx1/α − γx with f (x∗ ) given by Eq. (6), we obtain 0≥

f (dx∗ e) − f (x∗ ) f (x∗ + 1) − f (x∗ ) ≥ f (x∗ ) f (x∗ ) n o 1 1/α ∗ ∗ 1/α = · β · { + 1) − } − γ (x (x ) f (x∗ ) ) ( αα/(α−1) γ 1/(α−1) 1 ≥ · β· −γ 1−1/α β α/(α−1) α (x∗ + 1) β 1 αα/(α−1) γ 1/(α−1) ≥ · · −γ β α/(α−1) α (1 + (β/αγ)α/(α−1) )1−1/α αα/(α−1) γ 1/(α−1) β αγ = · · −γ β α/(α−1) α ((αγ)α/(α−1) + β α/(α−1) )1−1/α αα/(α−1) γ 1/(α−1) β αγ = · · −γ β α/(α−1) α αγ + β

(17) (18)

(19) (20)


=

13

−α(2α−1)/(α−1) γ α/(α−1) = −O (1 − ρ)α/(α−1) . α/(α−1) β (β + αγ)

Eq. (18) follows from Eq. (17) by the concavity of g(x) = x1/α function. Eq. (20) follows from Eq. (19) given that (a + b)1−1/α ≤ a1−1/α + b1−1/α . In the same way we can bound the quantity (f (bx∗ c) − f (x∗ )) /f (x∗ ). In summary, we have shown that the bounds presented in Theorem 2 is in fact almost tight.

3.2. Waiting Time in a Multi-Server Robust Queue We consider a queue with m parallel servers and denote by An the arrival time of the nth job where An =

Pn

`=1 T`

for every n, and Cn the completion time of the nth job, i.e., the time the nth job

leaves the system (including service). The central difficulty in analyzing probabilistic multi-server queues lies in the fact that overtaking may occur, i.e., the nth departing job is not necessarily the nth arriving job. To address this matter, we introduce the ordered sequence of completion times C(1) ≤ C(2) ≤ . . . ≤ C(n) and define Dn as the nth interdeparture time given by Dn = C(n) − C(n−1) . We briefly review the dynamics of the multi-server queue. Figure 1 depicts a two-server queue and the associated quantities of interest An , Xn , Wn , Cn , C(n) , and Dn .

Figure 1

Dynamics of a Two-Sever Queue.


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Looking at the snapshot of the process for five jobs, the waiting times can be found as W1 = 0, W2 = 0, W3 = C2 − A3 = C(1) − A3 , W4 = C1 − A4 = C(2) − A4 , W5 = C4 − A5 = C(3) − A5 . By induction, we obtain the general expression of the nth waiting time Wn = max{C(n−m) − An , 0}

(21)

as is well established (see Kiefer and Wolfowitz (1955)). Notice that Eq. (21) generalizes the Lindley recursion to the multi-server case. Furthermore, we note that Cn = An + Wn + Xn = An + Sn ,

(22)

C0 = 0 and Cr = Ar + Xr for 1 ≤ r ≤ m,

(23)

where Sn = Wn + Xn denotes the sojourn time of the nth job. Note that Eq. (23) follows by the c. fact that the first m jobs do not wait. We now provide a closed form expression for W s Theorem 3. If {Ti }i≥1 ∈ U a , {Xi }i≥1 ∈ Um , αa = αs = α and ρ < 1, then 1/(α−1) Γa + Γs /m1/α c ≤ α−1 · λ W 1/(α−1) αα/(α−1) (1 − ρ)

Proof of Theorem 3.

α/(α−1) .

(24)

By combining Eqs. (21), (22) we obtain

C(n−m) ≤ max C(n−2m) , An−m + Xn−m ≤ max max C(n−3m) , An−2m + Xn−2m , An−m + Xn−m , ≤ max C(n−3m) + Xn−2m + Xn−m , An−2m + Xn−2m + Xn−m , An−m + Xn−m . Let n = vm + r, 0 ≤ r < m, where r is the remainder of the division of n by m. Thus, ( ) v−1 v−1 X X C(n−m) ≤ max C(n−vm) + Xn−km , An−(v−1)m + Xn−km , . . . , An−m + Xn−m . k=1

k=1

From Eq. (21), the nth waiting time is as follows ( ) v−1 v−1 X X Wn ≤ max C(n−vm) + Xn−km − An , An−(v−1)m + Xn−km − An , . . . , An−m + Xn−m − An , 0 . k=1

k=1


15

Note that n − vm = r and Wr = 0 yielding C(r) ≤ Cr = Ar + Xr , for all 0 ≤ r < m. Then, ( ) v−1 v−1 X X Wn ≤ max Ar + Xr + X(v−k)m+r − An , Am+r + X(v−k)·m+r − An , . . . , An−m + Xn−m − An , 0 . k=1

k=1

Pn Expressing the arrival times as An = `=1 T` for every n, we obtain  v n v−1 n X X X X Wn ≤ max X(v−k)m+r − T` , X(v−k)m+r − T` , . . . , Xn−m −  k=1

`=r+1

k=1

`=m+r+1

n X `=(v−1)m+r+1

By substituting ` = v − k, the above expression can be re-written as ) ( v−1 vm+r X X T` , 0 . Wn ≤ max X`m+r − 0≤j≤v−1

  T` , 0 . 

(25)

`=jm+r+1

`=j

Note that if we let m = 1 in Eq. (25), we recover Eq. (7) for the single-server case. Also, note that the above bound is tight in the case where overtaking does not occur and jobs leave in the order s of their arrivals, i.e., C(i) = Ci , i ≥ 1. Since {Xi }i≥1 ∈ Um and {Ti }i≥1 ∈ U a v−1 X `=j

X`m+r ≤

v−j + Γs (v − j)1/α , µ

vm+r X

T` ≥

`=jm+r+1

m(v − j) − m1/α Γa (v − j)1/α , λ

(26)

we obtain cn ≤ max W

0≤j≤v−1

m

1/α

Γa + Γs (v − j)

1/α

m(1 − ρ) − (v − j) . λ

Making the change of variables x = v − j, we obtain that cn ≤ max W

1≤x≤v

m

1/α

1/α

Γa + Γs x

m(1 − ρ) − x , λ

leading to c ≤ max W 1≤x

m

1/α

1/α

Γa + Γs x

m(1 − ρ) − x . λ

Note that the right-hand side is an one dimensional concave maximization problem in the form of Eq. (12), maximized at x∗ (refer to Eq. (13)), with β = m1/α Γa + Γs and γ = m(1 − ρ)/λ > 0, given ρ < 1. Substituting β and γ by their respective values in Eq. (12) yields Eq. (24).

Remark on Theorem 3 - Tightness of the bound We had earlier shown that the bound in Theorem 2 is in fact tight. Here, we also present a sequence of interarrival times that exactly achieves a waiting time equal to the bound in Eq. (24). Consider

16


the following sequence of interarrival and service times, and assume x∗ is an integer given by x∗ = mv + r and let j ∗ = v − v,  1/λ    Ti = 1 Γa    − ∗ (α−1)/α λ (x )

Xim+r =

i = 1, . . . , mj ∗ + r, (27) i = mj ∗ + r + 1, . . . , n,

 1/µ     1 + Γs µ v (α−1)/α

i = 1, . . . , j ∗ − 1, (28) i = j ∗ , . . . , v − 1.

s As before, these sequences belong to U a and Um , respectively and also do not allow any overtaking.

To see that they do not allow overtaking, note that a later customer always experiences a worse service time and an early interarrival time leading to more waiting time. Along the same lines for the case of single server, these sequences lead to a waiting time that is exactly equal to the upper bound provided in Eq. (24) when x∗ is an integer, or it is within a factor of O (1 − ρ)α/(α−1) when x∗ is fractional.

3.3. Implications and Insights To summarize, we obtain the following characterization of the waiting time in an m-server queue 1/(α−1) Γa + Γs /m1/α c ≤ α−1 · λ W 1/(α−1) αα/(α−1) (1 − ρ)

α/(α−1) ,

(29)

where the expression simplifies to Eq. (6) for m = 1. We present next the main implications and insights that follow from this closed form expression. (a) Qualitative Insights The robust queue behaves qualitatively the same as the traditional queue. For instance, the classical i.i.d. arrival and service processes with finite variance can be modeled by setting α = 2. For the single server queue, Eq. (29) becomes 2

cn = λ · (Γa + Γs ) , W 4 (1 − ρ)

(30)


17

and for the multi-server queue 1/2 cn = λ · Γa + Γs /m W 4 1−ρ

2 .

(31)

Contrasting Kingman’s bounds (2) and (3) with the bounds (30) and (31), we observe that they have the same functional dependence on λ/(1 − ρ) and on the variability parameters Γ2a , Γ2s /m, (correspondingly σa2 , σs2 /m). In this sense, our approach leads to the same qualitative insights as stochastic queueing theory. (b) Modeling Heavy-Tailed Behavior Our approach allows a closed-form expression for the steady-state waiting time for all values of α ∈ (1, 2), which include heavy tailed random variables. Observe that heavier the tail, i.e., the smaller the tail coefficient α, the higher the order of the waiting time, given its dependence on 1/(1 − ρ)1/(α−1) . To illustrate, a decrease in the tail coefficient from α = 2 to α = 1.5 increases the waiting time by one order of magnitude. This is in agreement with the stochastic queueing theory literature, where it is known that the waiting time exhibits a heavy-tailed distribution under heavy tailed services (see Whitt (2000), Crovella (2000)). During the past decade, studies have shown the existence of heavy-tailed behavior in some of the main application areas of queueing theory: (a) Call centers (Barabasi (2005)), (b) Data centers and cloud computing (Loboz (2012), Benson et al. (2010)), and (c) Internet (Willinger et al. (1998), Leland et al. (1995), Crovella (1997), Jelenkovic et al. (1997), Kumar et al. (2000)). Our closed form results allow practioners in these application domains to understand the dependence of waiting times on various system parameters such as the number of servers m and traffic intensity ρ.

4. The Robust Burke’s Theorem Understanding the mechanism of the departure process from a queue is central to analyzing networks of queues. As we discussed in Section 1, renewal arrivals do not lead to renewal departures with the exception of the M/M/m queue. In contrast, our approach allows an exact characterization of the departure process by providing an analog to Burke’s theorem: If a single-server queue has arrivals belonging to the set U a , the departures also belong to the set U a .


18

4.1. Departure Process in a Single-Sever Robust Queue We consider the single-sever robust queue and characterize the uncertainty set of the departure process. Theorem 4. If {Ti }i≥1 ∈ U a , {Xi }i≥1 ∈ U s , αa = αs = α and ρ < 1, then the interdeparture times {Di }i≥1 belong to the uncertainty set   n X n − k       Di −     λ 1 i=k+1 d U = (D1 , D2 , . . . , Dn ) ≤ Γ + O , ∀ k ≤ n − 1 . a   (n − k)1/α (n − k)1/α        

Proof of Theorem 4.

(32)

The nth interdeparture time is expressed as Dn = Tn + Wn − Wn−1 + Xn −

Xn−1 = Tn + Sn − Sn−1 , implying n X

n X

Di =

i=k+1

Ti + Sn − Sk ,

(33)

i=k+1

where by Eq. (7) ` X

S` = max

1≤j≤`−1

!

` X

Xi −

i=j

Ti .

(34)

Ti + Sn .

(35)

i=j+1

Thus, the sum of interdeparture times can be written as n X

Ti − Sk ≤

i=k+1

n X

Di ≤

i=k+1

n X i=k+1

We seek to minimize the left-hand-side and maximize the right-hand side of Eq. (35) over sets U s and U a . By Assumption 1 and dividing through by (n − k)1/α , we obtain n X

−Γa −

Sbk ≤ (n − k)1/α

Di −

i=k+1

n−k λ ≤ Γa +

(n − k)1/α

Sbn , (n − k)1/α

(36)

where ( Sb` =

max

X∈U s ,T∈U a

max

1≤j≤`−1

= max

1≤j≤`−1

` X

Γa (` − j)

1/α

i=j

Xi −

` X

!) Ti

,

i=j+1 1/α

+ Γs (` − j + 1)

1−ρ − (` − j) λ

1 + . µ

(37)


19

The one-dimensional concave maximization problem in Eq. (37) is of the form max β · x1/α + δ (x + 1)

1≤x≤`−1

1/α

− γ · x ≤ max (β + δ)(x + 1)1/α − γ(x + 1) + γ, 1≤x≤`−1

α/(α−1)

≤

α − 1 (β + δ) αα/(α−1) γ 1/(α−1)

+ γ,

(38)

where β = Γa , δ = Γs , γ = (1 − ρ)/λ > 0, given ρ < 1. Note that the bound (38) is not tight unless α/(α−1)

` ≥ [(β + δ)/αγ]

. For k ≤ n − 1 where n is large, substituting β, δ and γ by their respective

values yields Sbk Sbn 1 ≤ ≤ 1/α 1/α (n − k) (n − k) (n − k)1/α

α/(α−1)

α − 1 λ1/(α−1) · (Γa + Γs ) · 1/(α−1) αα/(α−1) (1 − ρ)

1 + λ

!

=O

1 (n − k)1/α

Applying the above bounds to Eq. (36) completes the proof.

4.2. Departure Process in a Multi-Sever Robust Queue We generalize the characterization of the departure process uncertainty set to the multi-server robust queue. s Theorem 5. For an m-server queue, if {Ti }i≥1 ∈ U a , {Xi }i≥1 ∈ Um , αa = αs = α and ρ < 1 , then

the interdeparture times {Di }i≥1 belong to the uncertainty set   n X n − k       Di −     λ 1 i=k+1 d U = (D1 , D2 , . . . , Dn ) ≤ Γa + O , ∀k ≤ n − 1 .   (n − k)1/α (n − k)1/α         Proof of Theorem 5.

(39)

The sum of the interdepartures times can be expressed as n X

Di = C(n) − C(k) .

(40)

i=k+1

Given that the `th arrival cannot occur after the `th departure from the queue, i.e., A` ≤ C(`) , and knowing that C(`) ≤ C` = A` + S` (see Eqs. (22) and (??)), the sum of the interdeparture times can be bounded by An − Ak − Sk ≤

n X i=k+1

Di ≤ An − Ak + Sn .

.


20

Replacing An − Ak =

Pn

i=k+1 Ti ,

we recover Eqs. (35) and (36). We would like to determine the

bounds on the highest sojourn times Sbk and Sbn , similarly to the single-server case. Let ` = wm + s where r is the remainder of the division of ` by m. Thus, by Eq. (25) !) ( w wm+s X X , Sb` = max max Xim+s − Ti X∈U s ,T∈U a

0≤j≤w−1

i=j

i=jm+s+1

m(1 − ρ) 1 1/α 1/α = max Γa (w − j) + Γs (w − j + 1) − (w − j) + . 0≤j≤w−1 λ µ

(41)

Note that the maximization term in Eq. (41) is similar to that in Eq. (37). We can therefore apply Eq. (38) with β = m1/α Γa , δ = Γs and γ = m(1 − ρ)/λ. For k ≤ n − 1 where n is large, substituting β, δ and γ by their respective values yields Sbn 1 Sbk ≤ ≤ 1/α 1/α (n − k) (n − k) (n − k)1/α

α/(α−1)

α − 1 λ1/(α−1) · (Γa + Γs /m1α ) · 1/(α−1) αα/(α−1) (1 − ρ)

1 + λ

!

=O

1 (n − k)1/α

Applying the above bounds to Eq. (36) completes the proof.

4.3. Implications and Insights To summarize, we obtain the following characterization of the uncertainty set describing the departure process from a multi-server queue   n X n − k       Di −     λ 1 i=k+1 d ≤ Γ + O , ∀ k ≤ n − 1 U = (D1 , D2 , . . . , Dn ) . a   (n − k)1/α (n − k)1/α        

(42)

We next discuss the implications and insights that follow from the Robust Burke’s Theorems. (a) While the construction of the arrival uncertainty set using limit laws is motivated by iid interarrival times, membership in U a does not imply that the interarrival times are independent. Similarly, membership in U d does not imply that {D1 , . . . , Dn } are independent. This is consistent with stochastic queueing theory, where the interdeparture times are in general dependent. Our framework provides a concrete way to characterize the departure process. (b) Robust Burke’s Theorem: The contribution from the service process is included in the O 1/(n − k)1/α term. As the number of jobs n tends to infinity, this term tends to zero,

.


21

implying that the steady-state departure process can be characterized by the exact same uncertainty set describing the arrival process, i.e. {Di }i≥1 ∈ U a , for both single and multiserver queues. This result generalizes Burke’s theorem and is the cornerstone of our queueing network analysis covered in Section 5.

5. The Robust Queueing Network Analyzer Consider a network of J queues serving a single class of jobs. Each job enters the network through some queue j, and either leaves the network or departs towards another queue right after completion of his service. The primitive data in the queueing network are: (a) External arrival processes with parameters (λj , Γa,j , αa,j ) that arrive to each node j = 1, . . . , J. (b) Service processes with parameters (µj , Γs,j , αs,j ), and the number of servers mj , j = 1, . . . , J. (c) Routing matrix F = [fij ], i, j = 1, . . . , J, where fij denotes the fraction of jobs passing through queue i and are routed to queue j. The fraction of jobs leaving the network from queue i is P 1 − j fij . In order to analyze the waiting time in a particular queue j in the network, we need to characterize the overall arrival process to queue j and then apply Theorem 2 for single-server and Theorem 3 for multi-server queues. The arrival process in queue j is the superposition of different processes, each of which is either a process from the outside world, or a departure process from another queue, or a thinning of a departure process from another queue, or a thinning of an external arrival process. Correspondingly, in order to analyze the network, we need to characterize the effect that the following operations have on the arrival process: (a) Passing through a queue: Under this operation, we characterize the departure process {Di }i≥1 when an arrival process {Ti }i≥1 ∈ U a passes through a queue. We have already accom-

plished this in Theorems 4 and 5 for the single-server and multi-server queue, respectively, and have shown that {Di }i≥1 ∈ U a when n → ∞. (b) Superposition of arrival processes: Under this operation, m arrival processes {Tij }i≥1 ∈ Uja , j = 1, . . . , m combine to form a single arrival process. Theorem 6 characterizes the uncertainty set of the combined arrival process.


22

(c) Thinning of an arrival process: Under this operation, a fraction f of arrivals from a given arrival process is classified as type I while the remaining arrivals are classified as type II. In Theorem 7, we characterize the uncertainty set of the resulting thinned type I process.

5.1. The Superposition Process Let us consider a queue j that is fed by m arrival processes. Let Uja denote the uncertainty set representing the interarrival times {Tij }i≥1 from arrival process j = 1, . . . , m. We denote the uncera tainty set of the combined arrival process by Usup . Given the primitives (λj , Γa,j , α), j = 1, . . . , m,

we define the superposition operator n o (λsup , Γa,sup , αsup ) = Combine (λj , Γa,j , α) , j = 1, . . . , m ,

where (λsup , Γa,sup , αsup ) characterize the merged arrival process {Ti }i≥1 . Theorem 6 (Superposition Operator). The superposition of arrival processes characterized by the uncertainty sets n  X n − k   Ti − a λ Uj = (T1 , . . . , Tn ) i=k+1 , j = 1, . . . , m, j  ≤ Γa,j , ∀k ≤ n − 1    1/α (n − k)   

(43)

results in a merged arrival process characterized by the uncertainty set   n X n − k     Ti − a λ , Usup = (T1 , . . . , Tn ) i=k+1 sup  ≤ Γa,sup , ∀k ≤ n − 1    1/α (n − k) where m X

λsup =

m X j=1

λj , αsup = α, Γa,sup =

!(α−1)/α (λj Γa,j )

α/(α−1)

j=1 m X

·

(44)

λj

j=1

Proof of Theorem 6.

We first provide a proof for the case where m = 2, and then generalize the

result through induction.


23

(a) Let {Tij }i≥1 ∈ Uja , j = 1, 2 where Uja is given by Eq. (43), that is nj X

λj

Tij ≤ (nj − kj ) + λj Γa,j (nj − kj )

1/α

, j = 1, 2.

i=kj +1

Summing over index j = 1, 2, we obtain λ1

n1 X i=k1 +1

n2 X

Ti1 +λ2

Ti2 ≤ (n1 − k1 + n2 − k2 )+λ1 Γa,1 (n1 − k1 )

1/α

+λ2 Γa,2 (n2 − k2 )

1/α

. (45)

i=k2 +1

Without loss of generality, consider the time window T between the arrival of the k1th and the nth 1 jobs from arrival process 1, and assume that, within period T , the queue sees arrivals of jobs (k2 + 1) up to (n2 − 1) from arrival process 2, n1 X

T=

Ti1

i=k1 +1

n2 X

≤

Ti2 .

(46)

i=k2 +1

During time window T , the queue receives a total of (n1 − k1 + n2 − k2 ) jobs, with (n1 − k1 + 1) arrivals detected from the first arrival process (including job k1 ), and (n2 − k2 − 1) arrivals from second arrival process. Therefore, period T can also be written in terms of the combined interarrival times {Ti }i≥1 as n X

T=

Ti , where k = k1 + k2 , and n = n1 + n2 .

(47)

i=k+1

Combining Eqs. (46) and (47) yields (λ1 + λ2 )

n X i=k+1

n1 X

Ti ≤ λ1

Ti1 + λ2

i=k1 +1

n2 X

Ti2

i=k2 +1

which by Eq. (45) can be written as (λ1 + λ2 )

n X

Ti ≤ (n − k) + λ1 Γa,1 (n1 − k1 )

1/α

+ λ2 Γa,2 (n2 − k2 )

1/α

.

(48)

i=k+1

Rearranging and dividing both sides by (λ1 + λ2 ) and (n − k) n X

1/α

, we ontain

n−k λsup

Ti −

i=k+1

1/α

(n − k)

λ1 Γa,sup (n, k) = Γa,1 λ1 + λ2

≤ Γa,sup (n, k), where λsup = λ1 + λ2 , αsup = α, and

n1 − k1 n1 − k1 + n2 − k2

1/α

λ2 + Γa,2 λ1 + λ2

n2 − k2 n1 − k1 + n2 − k2

1/α .


24

By letting x=

n1 − k1 , n1 − k1 + n2 − k2

(49)

the maximum value that Γa,sup (n, k) can achieve over the range of (n, k) can be determined by optimizing the following one-dimensional concave maximization problem over x ∈ (0, 1) max

n o (α−1)/α 1/α βx1/α + δ (1 − x) = β α/(α−1) + δ α/(α−1) ,

(50)

x∈(0,1)

where β=

λ2 λ1 Γa,1 , and δ = Γa,2 . λ1 + λ2 λ1 + λ2

Substituting β and δ by their respective values in Eq. (50) completes the proof for m = 2 with h i(α−1)/α α/(α−1) α/(α−1) + (λ2 Γa,2 ) (λ1 Γa,1 ) Γa,sup = . λ1 + λ2 We refer to this procedure of combining two arrival processes by the operator (λsup , Γa,sup , αsup ) = Combine {(λ1 , Γa,1 , α) , (λ2 , Γa,2 , α)} . (b) Suppose that the arrivals to a queue come from arrival processes 1 through (m − 1). We assume that the combined arrival process belongs to the proposed uncertainty set, with !(α−1)/α m−1 X α/(α−1) (λj Γa,j ) m−1 X j=1 λ= λj and Γa = · λ j=1 Extending the proof to m sources can be easily done by repeating the procedure shown in part (a) through the operator (λsup , Γa,sup , αsup ) = Combine λ, Γa , α , (λm , Γa,m , α) . 5.2. The Thinning Process We consider an arrival process {Ti }i≥1 in which a fraction f of arrivals are classified as type I and the remaining are classified as type II. Given the primitives (λ, Γa ) of the original process and the fraction f , we define the thinning operator n

(λsplit , Γa,split , α) = Split (λ, Γa , α) , f

o

where (λsplit , Γa,split , α) characterizes the thinned arrival process {Tisplit }i≥1 .


25

Theorem 7 (Thinning Operator). The thinned arrival process of a fraction f of arrivals belonging to U a is described by the uncertainty set  n  X split n − k     Ti − split a split λsplit Usplit = (T1 , . . . , Tn ) i=k+1 ,  ≤ Γa,split , ∀k ≤ n − 1    1/α (n − k)

(51)

1/α 1 . where λsplit = λ · f and Γa,split = Γa · f Proof of Theorem 7.

Consider an arrival process described by U a and consider the time window

between the k th and the nth arrivals. Suppose that a fraction f of these arrivals are type I arrivals, i.e. , out of the total of (n − k) arrivals excluding the k th job, (nsplit − ksplit ) are type I arrivals, such that f=

nsplit − ksplit . n−k

Let {Tisplit }i≥1 denote the interarrival times in the thinned arrival process. Note that nsplit

X

Tisplit ≤

i=ksplit +1

n X

Ti ,

i=k+1

with equality satisfied when the k th and nth jobs are both classified as type I. By Assumption 1, we obtain nsplit

X

Tisplit ≤

i=ksplit +1

n−k 1/α + (n − k) Γa . λ

We obtain the upper bound in Eq. (51) by substituting (n − k) by (nsplit − ksplit )/f . The lower bound is derived similarly, hence completing the proof.

Remark: The superposition and thinning operators are consistent. In fact, it is easy to check that, for splitting fractions fj such that

Pm

j=1 fj

= 1,

n o Combine Split (λ, Γa , α) , f j , j = 1, . . . , m = (λ, Γa , α) . 5.3. The Overall Network Characterization Having built an understanding of the superposition, departure and thinning processes, we are now in a position to present our approach for performing exact analysis of queueing networks, which is a


26

major contribution of this paper. We perceive the queueing network as a collection of independent queues that could be analyzed separately. We employ the Combine and Split operators in view of characterizing the effective arrival process to each queue in the network. Knowledge of the effective arrival process allows to study the jobs’ waiting time at this queue through Theorems 2 and 3 as well as its departure process through Theorems 4 and 5 for a single-server and multi-sever queue, respectively. Theorem 8 characterizes the effective arrival process perceived at each queue in the network. Theorem 8. The behavior of a single class queueing network is equivalent to that of a collection of independent queues, with the arrival process to node j characterized by the uncertainty set   n X j n − k     Ti − j a j λ Uj = (T1 , . . . , Tn ) i=k+1 , j = 1 . . . , J, j  ≤ Γa,j , ∀k ≤ n − 1    1/α (n − k) where

λ1 , λ2 , . . . , λJ and Γa,1 , Γa,2 , . . . , Γa,j satisfy the set of equations for all j = 1, . . . , J

λj = λj +

J X

λi f ij ,

(52)

i=1

" (λj · Γa,j ) Γa,j =

Proof of Theorem 8.

α/(α−1)

+

J X

λi · Γa,i

α/(α−1)

#(α−1)/α · f ij

i=1

λj

.

(53)

Let us consider a queue j receiving jobs from

(a) external arrivals described by parameters (λj , Γa,j , α), and (b) internal arrivals routed from queues i, i = 1, . . . , J resulting from splitting the effective departure process from queue i by f ij . By Theorems 4 and 5, the effective departure process from queue i has the same form as the effective arrival process to queue i described by the parameters λi , Γa,i , α . The effective arrival process to queue j can therefore be represented as n o λj , Γa,j , α = Combine (λj , Γa,j , α) , Split λi , Γa,i , α , f ij , i = 1, . . . , J

(54)


27

By Theorem 7, we substitute the split processes by their resulting parameters and obtain the superposition of J + 1 arrival processes 1/α ! 1 , α , i = 1, . . . , J λj , Γa,j , α = Combine (λj , Γa,j , α) , f ij λi , Γa,i f ij

Applying now Theorem 6 yields Eqs. (52) and (53).

(55)

Note that in our analysis, we have assumed that each queue in the network perceives one stream of external arrivals. However, Theorem 8 can be extended in the case where external arrivals are thinned among different queues in the network. This can be done by adding a node in the network for each thinned external arrival process and appending its thinning probabilities to the transition matrix F. We now provide the main insights and implications that arise from Theorem 8. (a) Network Performance Analysis: Theorem 8 allows us to compute performance measures cnj in a queueing network by considering the queues separately. For instance, the waiting time W at queue j can be determined through Theorems 2 and 3 with an effective arrival parameters (λj , Γa,j , α) and service parameters (µ, Γs , α). (b) Tractable System Solution: Determining the overall network parameters (λ, Γ) amounts to solving a set of linear equations. To see this, substitute xj = λj Γa,j

α/(α−1)

, for all j = 1, . . . , J,

in Eqs. (52) and (53) to obtain the following linear system of equations  J X    λ = λ + λi f ij j = 1, . . . , J,  j j    i=1  J  X   α/(α−1)  x = Γ + f ij xi ) (λ  j j a,j 

j = 1, . . . , J.

i=1

Given that the routing matrix F = {fij } is sub-stochastic, the linear system of equations solves for (λj , xj ), hence allowing to determine Γa,j , for all j = 1, . . . , J.

6. Queues with Asymmetric Heavy-tailed Arrival and Service Processes In this section, we extend our results in Sections 3-5 for the case of asymmetric heavy-tailed arrival and service processes, that is, when αa 6= αs . We present analogs of Theorems 2-8 that allow us

28


to analyze queueing networks composed of queues with arbitrary values for αa ’s and αs ’s. In this section, we let 1{x=y} denote the indicator variable defined by ( 1, if x = y, 1{x=y} = 0, otherwise. We omit the proofs as they are straightforward generalizations of the proofs in Theorems 2-8. Waiting Times with αa 6= αs c by solving the optimization problem As discussed in Section 3, we compute the waiting times W max βx1/αa + δx1/αs − γx,

(56)

x

where β = m1/αa Γa , δ = Γs , and γ = m(1 − ρ)/λ· Problem (56) is a concave maximization problem, which can be solved numerically using Newton’s method. We next present a bound on the waiting time, which is asymptotically tight as ρ → 1, but not necessarily tight for other ρ. s Theorem 9. If {Ti }i≥1 ∈ U a , {Xi }i≥1 ∈ Um and ρ < 1, then 1/(α−1) Γa + Γs /m1/α c ≤ α−1 · λ W 1/(α−1) αα/(α−1) (1 − ρ)

α/(α−1) ,

where α = min (αa , αs ). Note that the exponent of (1 − ρ) depends on the α term corresponding to the heaviest among the arrival and service processes. Departure Processes with αa 6= αs The Robust Burke Theorem generalizes to the case of asymmetric tails with the departure uncertainty set described as in Eq. (42) with tail coefficient αa . Note that the contribution of the service process is included in the O 1/(n − k)1/αa term.

Superposition Process In Section 5, we have assumed that all the combining arrival processes have the same value for αa = α. We next consider the case when the arrival streams are characterized by different α0 s.


29

Theorem 10. The superposition of arrival processes characterized by the uncertainty sets   n X n − k     Ti − a λj Uj = (T1 , . . . , Tn ) i=k+1 , j = 1, . . . , m,  ≤ Γa,j , ∀k ≤ n − 1    1/αa,j (n − k)

(57)

results in a merged arrival process characterized by the uncertainty set   n X n − k     Ti − a λ , Usup = (T1 , . . . , Tn ) i=k+1 sup  ≤ Γa,sup , ∀k ≤ n − 1    1/α (n − k) where m X

λsup =

m X

λj , α = min αa,j , and Γa,sup =

1{αa,j

= α}

· (λj Γa,j )

j=1

j

j=1

!(α−1)/α αa,j /(αa,j −1)

m X

·

λj

j=1

Thinning Process We note that the split arrival process inherits the αa term corresponding to the thinned arrival process. Hence, Theorem 7 still holds in this case. The Generalized Queueing Network We consider a queueing network and characterize the parameters of the effective arrival processes to each queueing node in the network under the assumption of asymmetric tail behavior. We observe that the parameter αa,j describing the tail behavior of the effective arrival process depends on the tail behavior of all the queueing nodes that communicate with node j. Theorem 11. Consider a queueing network with J queues and external arrival processes characterized by (λj , Γa,j , αa,j ). The behavior of this network is equivalent to that of a collection of independent queues, with the arrival process to node j characterized by the uncertainty set  n  X n − k     Ti − a λj , j = 1 . . . , J, U j = (T1 , . . . , Tn ) i=k+1   ≤ Γ , ∀ k ≤ n − 1   a,j 1/α (n − k) a,j where λ1 , λ2 , . . . , λJ and Γa,1 , Γa,2 , . . . , Γa,J satisfy the set of equations for all j = 1, . . . , J λj = λj +

J X i=1

λi f ij ,


30

" 1{αa,j = αa,j } · (λj · Γa,j )

αa,j /(αa,j −1)

Γa,j =

+

J X

1{αa,i = αa,j } · λi · Γa,i

αa,i /(αa,i −1)

#(αa,j −1)/αa,j · f ij

i=1

λj

,

with αa,j = mini:{i→j} αa,i , where {i → j } means that node i communicates with node j in the network with routing matrix F. We provide the main insights and implications that arise from Theorem 11. (a) Effect of Heavier Tails: Theorem 11 implies that the tail behavior of the effective arrival process at a given queue is determined by the “heaviest” tail among all departure processes arriving to this queue including the external arrival process to the queue. If all nodes communicate with each other, the tail behavior of the queueing network is then determined by the heaviest tail among the external arrival processes. (b) Tractable System Solution: Note that the set of equations that characterize the effective arrival process are similar to Eqs. (52) and (53). The only difference in the system for the asymmetric case is the presence of indicator variables 1{αa,j = αa,j } which isolate the heaviest tail among the merged arrival processes at any given node. Given that these indicator variables are known from data, one could think of this system as a linear system of equations (as for Eqs. (52) and (53)) with f˜ij = 1{αa,i = αa,j } · fij . The modified routing matrix with entries f˜ij remains sub-stochastic allowing a unique solution to this system of linear equations.

7. Computational Results In this section, we present computational results to demonstrate the effectiveness of our approach in analyzing queueing networks. We shall refer to our approach as the Robust Queueing Network Analyzer (RQNA) in the remainder of this paper. Our objectives consist of (1) comparing the results obtained by RQNA with the results obtained from simulation and the Queueing Network Analyzer (QNA) proposed by Whitt (1983), and (2) investigating the relative performance of RQNA with respect to system’s network size, degree of feedback, maximum traffic intensity, and diversity of external arrival distributions.


31

In view of comparing our approach to simulation and QNA, we consider instances of stochastic queueing networks with the following primitive data: (a) The distributions of the external arrival processes with parameters (λj , σa,j , αa,j ) with coeffi2 cients of variation c2a,j = λ2j σa,j , j = 1, . . . , J.

(b) The distributions of the service processes with parameters (µj , σs,j , αs,j ) with coefficients of 2 variation c2s,j = µ2j σs,j and the number of servers mj , j = 1, . . . , J.

(c) Routing matrix F = [fij ], i, j = 1, . . . , J, where fij denotes the fraction of jobs passing through P queue i routed queue j. The fraction of jobs leaving the network from queue i is 1 − j fij . To apply RQNA on stochastic queueing networks, we first need to translate the stochastic primitive data given above into robust primitive data, namely uncertainty sets with appropriate variability parameters (Γa,j , Γs,j ) for each j = 1, . . . , J. To achieve this goal, we first describe in Section 7.1 how we use simulation on a single isolated queue to construct parameters (Γa , Γs ) given arrival and service distributions. This enables us to transform the stochastic data into uncertainty sets over external arrival and service processes. We then present in Section 7.2 an algorithm that details the procedure employed by RQNA to compute the desired performance measures for a network of queues. In Section 7.3, we report on the performance of RQNA in comparison to QNA and simulation, while in Section 7.4, we discuss on the performance of RQNA as a function of various network parameters. 7.1. Derived Variability Parameters Along the lines of QNA, we use simulation to construct appropriate functions for the variability parameters. To do so, we consider a single queue with m servers characterized by (ρ, σa , σs , αa , αs ) and model its variability parameters (Γa , Γs ) as follows Γa = σa and Γs = f (ρ, σa , σs , αa , αs ).

(58)

Motivated by Kingman’s bound (see Eq. (3)), we consider the following functional form for f (·) f (ρ, σs , σa , αa , αs ) = θ0 + θ1 · σs2 /m + θ2 · σa2 ρ2 m

(α−1)/α

− σa m(α−1)/α ,


32

where α = min{αa , αs }. We run simulation over multiple instances of a single queue while varying parameters (ρ, σa , σs , αa , αs ) for different arrival and service distributions. We employ linear cn regression to generate appropriate values for θ0 , θ1 and θ2 such that the values obtained for W by Theorem 9 are adapted according to the expected values of the waiting time obtained from simulation. We propose two different adaptation regimes: (a) Service Distribution Dependent Adaptation Regime where we allow the set of values (θ0 , θ1 , θ2 ) to depend on the service distribution, and (b) Service Distribution Independent Adaptation Regime where we obtain a single set of values (θ0 , θ1 , θ2 ). We would naturally expect that knowledge of the specific service distributions leads to more accurate answers and indeed this is verified by the results below. The motivation for considering the service independent adaptation regime is that often we might not know the service time distributions. We also note that we do not perform an adaptation of the values of (θ0 , θ1 , θ2 ) for each arrival distribution, since in the network, we have no prior knowledge of the arrival distribution at a given queue. The only known distribution at each queue is in fact the service distribution, hence the proposed adaptation methods. We report the values obtained for both adaptation regimes in Table 1. Using the tabulated values for (θ0 , θ1 , θ2 ), Tables 2, 3, and 4 present the percentage errors Table 1 (θ0 , θ1 , θ2 )

Service Adaptation Regimes

Service Dependent

Service Independent

Pareto

Normal

θ0

-0.05

-0.02

-0.063

θ1

1.09

1.0301

1.072

θ2

1.11

1.0409

1.068

of the waiting times obtained by Theorem 9 relative to the expected waiting time values obtained by simulation for a single queue with multiple servers. We observe that errors are within 8% and 9.5% of simulation for the single-server and multi-server queue, respectively. Note that adapting for the different service distributions leads to smaller errors


Table 2

33

Single-Server Queue: Waiting time percent errors relative to simulation.

Case

Pareto Distribution

Normal Distribution

(c2a , c2s )

RQNA∗

RQNA∗∗

RQNA∗

(0.25,0)

6.812

2.912

6.547

1.185

(0.25,1)

-7.314

-3.080

6.483

3.246

(0.25,4)

-6.583

-2.708

-7.268

-3.172

(1,0)

7.665

0.982

-5.505

-2.960

(1,1)

6.423

1.357

-6.579

-2.114

(1,4)

6.123

2.485

6.070

-1.532

(4,0)

-5.701

-1.729

6.050

1.380

(4,1)

-7.839

-1.824

-7.434

-3.084

(4,4)

-5.886

-2.231

7.405

1.334

∗

Table 3

Service Independent

∗∗

RQNA∗∗

Service Dependent

Multi-Server Single Queue: Waiting time percent errors ( Service Independent).

Case

3 servers

6 servers

10 servers

(c2a , c2s )

Normal

Pareto

Normal

Pareto

Normal

(0.25, 0)

8.822

-9.356

8.820

-9.599

8.453

-9.269

(0.25, 1)

7.238

-7.957

8.673

-9.159

9.264

-8.839

(0.25, 4)

-8.195

9.856

-8.878

8.255

-8.842

8.894

(1, 0)

-10.360

-8.833

-9.199

-8.698

-8.508

-9.420

(1, 1)

-7.985

8.929

-9.045

7.959

-8.583

9.365

(1, 4)

8.548

8.255

8.634

8.662

7.978

9.046

(4, 0)

-7.296

7.865

-8.609

8.485

-9.209

9.514

(4, 1)

-7.593

-7.913

-9.312

-9.010

-8.739

-9.553

(4, 4)

-9.372

-8.776

-9.028

-9.358

-8.960

-9.338

Table 4

Pareto

Multi-Server Single Queue: Waiting time percent errors in the (Service Dependent).

Case

3 servers

6 servers

10 servers

(c2a , c2s )

Normal

Pareto

Normal

Pareto

Normal

(0.25, 0)

6.031

-6.270

6.268

-7.404

6.448

-6.878

(0.25, 1)

5.440

-4.902

6.339

-6.582

7.380

-5.689

(0.25, 4)

-5.463

6.306

-5.966

5.185

-5.894

6.504

(1, 1)

-5.750

7.253

-6.155

6.035

-6.561

7.217

(1, 4)

6.655

6.024

6.909

6.036

6.869

7.145

(4, 0)

-5.714

5.678

-6.498

5.341

-7.483

7.689

(4, 1)

-6.028

-4.073

-6.112

-7.764

-6.522

-7.117

(4, 4)

-7.398

-7.818

-6.217

-5.810

-6.099

-7.183

Pareto

34


in comparison to the service independent adaptation regime, with errors within 3% and 7.5% for the single-server and multi-server queue, respectively.

7.2. The RQNA Algorithm Having derived the required primitive data for our robust approach, we next describe the RQNA algorithm we employ to compute performance measures of a given network of queues.

ALGORITHM 1. Robust Queueing Network Analyzer Input: External arrival parameters (λj , σa,j , αa,j ), service parameters (µj , σs,j , αs,j ), and routing matrix F = [f ij ], for i, j = 1, . . . , J. Input also the service times distributions for the case of service dependent adaptation regime. c at each node j, j = 1, . . . , J. Output: Waiting times W 1. For each external arrival process i in the network, set Γa,i = σa,i . 2. For each queue j in the network with parameters (µj , σs,j , αs,j ), compute (a) the effective parameters λj , Γa,j , αa,j according to Theorem 11 and set ρj = λj /µj , (b) the variability parameter Γs,j = f ρj , Γa,j , σs,j , αa,j , αs,j , and c at node j using Theorem 9. (c) the waiting time W 3. Compute the total sojourn time of the network by computing (a) the set of all possible paths P in the network, (b) the fraction fp of jobs routed through each path p ∈ P , (c) the corresponding total sojourn time Sbp across each path p ∈ P by summing the individual waiting times and service times at all nodes associated with this path, P (d) the total sojourn time in the network Sb = p∈P fp Sbp .

Note that, in Step 2(b), we treat each queue j in the network separately as a single isolated queue with an effective arrival process described by the variability parameter Γa,j . Note that we use Γa,j


35

as an input to f (.) in place of the standard deviation. This is motivated from our use of Γa = σa for the single queue case (see Eq. (58)). It is also possible to compute Γs,j using either the service independent or the service dependent adaptation regime based on whether we know the specific service time distribution at each queue.

7.3. Performance of RQNA in Comparison to QNA and Simulation We consider the network shown in Figure 2 and perform computations assuming queues have either single or multiple servers, with normal or Pareto distributed service times. Table 5 reports the percentage errors between the expected sojourn times calculated by simulation and those obtained by each of QNA and RQNA, assuming all nine queues in the network have a single server. Note that the sojourn time is defined as the time elapsed between the arrival of a job to the network until his departure from the network.

Figure 2

The Kuehn’s Network (see Kuehn (1979)).

Tables 6 and 7 summarize the percentage errors for RQNA relative to simulation for queues with 3, 6, and 10 servers using the service independent and service dependent adaptation regimes, respectively. We observe that (a) RQNA produces results that are often significantly closer to simulated values compared to QNA. Improvements generally range one order of magnitude better in favor of RQNA.


36

(b) RQNA is fairly insensitive to the heavy-tailed nature of the service distributions. In fact, the sojourn time percentage errors for both the Pareto and normally distributed services are within the same order. (c) Adaptation of (θ0 , θ1 , θ2 ) to service distribution yields smaller errors up to 6%. (d) RQNA’s performance is generally stable with respect to the number of servers at each queue, yielding errors within the same range for instances with 3 to 10 servers per queue.

Table 5

Single-Server Network: Sojourn time percent errors relative to simulation.

Case

Pareto Distribution

RQNA∗

3.291

15.28

7.786

1.389

-8.82

-3.478

12.08

8.329

3.869

20.13

-7.122

-3.052

11.57

-7.922

-3.882

(1, 0)

19.01

8.176

1.056

12.68

-6.367

-3.797

(1, 1)

14.06

6.832

1.799

5.84

-7.125

-2.555

(1, 4)

10.15

6.878

2.893

-10.45

7.911

-0.681

(4, 0)

21.82

-7.244

-1.934

10.95

6.739

1.290

(4, 1)

23.71

-8.729

-2.139

14.18

-9.28

-3.508

(4, 4)

17.51

-7.173

-2.974

11.55

9.251

1.671

QNA

RQNA

(0.25, 0)

22.78

8.281

(0.25, 1)

18.48

(0.25, 4)

∗

Table 6

∗

Normal Distribution QNA

(c2a,j , c2s,j )

RQNA

∗∗

Service Independent

∗∗

RQNA∗∗

Service Dependent

Multi-Server Network: Sojourn time percent errors (Service Independent).

Case

3 servers

6 servers

10 servers

(c2a,j , c2s,j )

Normal

(0.25, 0)

6.885

-7.212

7.328

-9.425

7.844

-7.995

(0.25, 1)

6.174

-5.815

7.724

-7.185

8.482

-6.872

(0.25, 4)

-6.327

7.386

-7.532

6.283

-7.138

7.823

(1, 0)

-6.184

-7.323

-6.926

-7.318

-7.174

-6.971

(1, 1)

-7.263

8.023

-6.982

7.162

-7.975

9.220

(1, 4)

7.382

6.931

7.373

6.942

8.207

10.908

(4, 0)

-6.862

7.418

-7.263

6.758

-8.219

9.710

(4, 1)

-6.598

-5.012

-7.566

-8.990

-8.458

-9.337

(4, 4)

-8.144

-8.404

-7.529

-6.845

-7.493

-8.874

Pareto

Normal

Pareto

Normal

Pareto


Table 7

37

Multi-Server Network: Sojourn time percent errors (Service Dependent).

Case

3 servers

6 servers

10 servers

(c2a,j , c2s,j )

Normal

(0.25, 0)

2.095

-2.732

2.628

-3.475

2.844

-3.655

(0.25, 1)

3.255

-0.803

4.034

-1.065

4.419

-0.992

(0.25, 4)

-2.067

1.416

-2.557

1.793

-2.760

1.911

(1, 0)

-2.254

-3.663

-2.886

-4.609

-2.877

-4.731

(1, 1)

-3.183

1.725

-4.133

2.232

-3.975

2.230

(1, 4)

3.859

1.529

4.978

1.936

5.118

1.998

(4, 0)

-3.852

4.628

-5.823

5.358

-5.429

5.309

(4, 1)

-3.272

-4.283

-4.372

-4.83

-4.228

-5.667

(4, 4)

-3.282

-4.123

-5.823

-5.823

-5.834

-6.129

Pareto

Normal

Pareto

Normal

Pareto

7.4. Performance of RQNA as a Function of Network Parameters We investigate the performance of RQNA (for the service dependent adaptation regime) as a function of the system’s parameters (network size, degree of feedback, maximum traffic intensity among all queues and number of distinct distributions for the external arrival processes) in families of randomly generated queueing networks. Tables 8 and 9 report the sojourn time percentage errors of RQNA relative to simulation as a function of the size of the network and the degree of feedback for queues with single and multiple servers, respectively. In the case of multi-server queueing networks, we randomly assign 3, 6 or 10 servers to each of the queues in the network independently of each other.

Table 8

Single-Server Networks: RQNA percent error as a function of network size and degree of feedback.

% Feedback loops / No of nodes

10

15

20

25

30

Feed-forward networks 0%

2.86

2.94

3.03

2.92

3.21

20%

3.12

3.25

3.29

3.71

3.64

35%

3.74

3.81

4.02

4.07

4.14

50%

4.42

4.63

4.84

5.23

5.65

70%

4.85

5.16

5.34

5.68

5.86

Tables 10 and 11 present the sojourn time percentage errors for RQNA relative to simulation as


38 Table 9

Multi-Server Networks: RQNA percent error as a function of network size and degree of feedback.

% Feedback loops / No of nodes

10

15

20

25

30

Feed-forward networks 0%

3.594

3.546

3.756

3.432

3.846

20%

3.696

4.014

4.02

4.392

4.452

35%

4.32

4.776

4.956

5.034

4.878

50%

4.95

4.806

5.358

5.67

6.192

70%

5.016

5.556

5.934

5.958

6.03

a function of the maximum traffic intensity among all queues in the network and the number of distinct distributions for the external arrival processes. Table 10 presents the results for networks with only single server queues, while Table 11 presents the results for networks in which each queue was randomly assigned 3, 6 or 10 servers. Specifically, we design four sets of experiments in which we use one type (normal), two types (Pareto and normal), three types (Pareto, normal and Erlang) and four types (Pareto, normal, Erlang and exponential) of arrival distributions. We observe that (a) Errors are slightly higher for multi-server networks compared to single-server networks. (b) RQNA’s performance is generally stable for higher degrees of feedback with errors below 6.2%. (c) RQNA is fairly insensitive to network size with a very slight increase in percent errors between 10-node and 30-node networks. (d) RQNA presents slightly improved results for lower traffic intensity levels. It is nevertheless fairly stable with respect to higher traffic intensity levels. (e) The percentage errors generally increase with diversity of external arrival distributions, but still are below 8.5% relative to simulation. Table 10

Single-Server Networks: RQNA percent error as a function of traffic intensity and variety of external arrival distributions.

No of different distributions

ρ = 0.95

ρ = 0.9

ρ = 0.8

ρ = 0.65

ρ = 0.5

1

3.34

3.26

3.17

3.02

2.72

2

6.38

5.85

5.47

4.87

3.24

3

7.43

7.09

6.04

5.88

4.53

4

7.56

6.98

6.81

6.29

5.18


Table 11

39

Multi-Server Networks: RQNA percent error as a function of traffic intensity and variety of external arrival distributions.

No of different distributions

ρ = 0.95

ρ = 0.9

ρ = 0.8

ρ = 0.65

ρ = 0.5

1

4.05

4.092

3.618

3.678

3.228

2

5.082

7.104

6.42

6.108

3.714

3

5.916

6.318

6.9

7.344

5.676

4

7.672

8.644

7.284

6.852

5.37

8. Concluding Remarks We revisited in this paper the problem of analyzing the performance measures of a single-class queue with multiple servers. While the analysis of the GI/GI/m queue is still an open problem under traditional queueing theory, we proposed an uncertainty set model which allows to solve it exactly. In particular, we derive a closed form expression for the waiting times and extend the analysis to arbitrary networks of queues through the following key principle: a) the departure from a queue, b) the superposition, and c) the thinning of arrival processes have the same uncertainty set representation as the original arrival processes. Our robust model also tackles heavy-tailed arrival and service processes, yielding closed-form solutions that are not available under traditional queueing theory. We proposed RQNA to analyze queueing networks and found that RQNA (with service dependent adaptation regime) yields results with error percentages in single digits (for all experiments we performed) relative to simulation and performs significantly better than QNA. Moreover, the performance of RQNA is to a large extent insensitive to the number of servers per queue, network size, degree of feedback and traffic intensity, and somewhat sensitive to the degree of diversity of external arrival distributions in the network. We feel that the proposed approach allows us to analyze queueing systems in a tractable way and fulfill the need to obtain both qualitative insights as well numerical tractability that has eluded queueing theory to a large extent in its over 100 years history. It also opens the door to analyze more involved queueing systems. Indeed we have extended the proposed approach in two directions: performance analysis of multi-class queueing networks in Bandi et al. (2012a) and performance analysis of queueing systems in the transient domain in Bandi et al. (2012b).


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