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Efficient Nonlinear Optimizations of
Queuing Systems
Mung Chiang, Arak Sutivong, and Stephen Boyd
Electrical Engineering Department, Stanford University, CA 94305
Abstract— We present a systematic treatment of efficient non- II. C ONVEX O PTIMIZATION AND G EOMETRIC
linear optimizations of queuing systems. The suite of formula- P ROGRAMMING
tions uses the computational tool of convex optimization, with
fast polynomial time algorithms to obtain the global optimum Convex optimization refers to minimizing a convex objective
for these nonlinear problems under various constraints. We first
function subject to upper bound inequalities on convex con-
show convexity structures of several queuing systems, including
some surprising transition patterns, followed by formulating and straint functions. The objective function can be generalized to
showing numerical examples of several convex performance op- be vector-valued, where the minimization is with respect to a
timizations for both single queues and queuing networks. Block- convex cone. These convex multiple-objective optimizations
ing probability minimization and service rate allocation through are useful for tradeoff analysis, and the notion of optimality
the effective bandwidth approach is also presented.
now becomes Pareto optimality [1].
Convex optimization problems can be easy to solve, both in
theory and in practice. Theoretically, showing an optimiza-
I. I NTRODUCTION
tion problem to be a strictly convex problem proves that there
Queuing systems form a fundamental part for different types is a unique global optimal solution, and leads to performance
of networks, including computer multiprocessor networks and bounds and sensitivity analysis through the dual problem.
communications data networks. Queuing systems are also an Practically, when put in the right form, convex optimization
integral part of various network elements, such as the input can be globally solved by fast polynomial time algorithms
and output buffers of a packet switch. We often would like to [9]. It also gives a good starting point to develop even simpler
optimize some performance metrics of queuing systems, for heuristics and establishes the optimal benchmark to compare
example, buffer occupancy, overall delay, jittering, workload, heuristics with.
and probabilities of certain states. In a network of queues, There is a particular type of convex optimization used in sec-
we may also have multiple conflicting objectives that need tions III, IV and V called geometric programming [1], [4],
to be optimally balanced. However, optimizing the perfor- which has also been applied to solve other network resource
mance of even simple queues like the M/M/m/m queue is allocation problems [6]. First, we have
in general a difficult problem because of the nonlinearity of Definition 1: A monomial is a function f : Rn → R, where
the performance metrics as functions of the arrival and ser- the domain contains all real vectors with positive components,
vice rates. Nonlinear optimization in general takes running and constants c ≥ 0, ai ∈ R:
time that scales exponentially with the problem size.
We show how convexity properties of queuing systems can be
used to turn some of these intractable problems into polyno- f (x) = cxa1 xa2 · · · xan .
1 2 n (1)
mial time solvable ones. By using the tool of convex optimiza- Definition 2: A posynomial is a sum of monomials f (x) =
tion, and in particular, geometric programming, we provide a a1k a2k ank
k ck x1 x2 · · · xn .
suite of formulations to efficiently optimize the performance
Geometric programming is an optimization problem in the
of queuing systems under Quality of Service (QoS) and fair-
following form:
ness constraints, first for single queues in section III, then for
blocking probability minimization and service rate allocation
minimize f0 (x)
through the effective bandwidth approach in section IV, then
subject to fi (x) ≤ 1, (2)
for networks of queues in section V, and for optimal feedback
hj (x) = 1
control in simple queuing networks in section V. The distin-
guishing characteristics of the formulations in this paper is where f0 and fi are posynomials and hj are monomials. Geo-
that they are nonlinear problems that can be solved as easily metric programming in the above form is not a convex op-
as linear problems by using convex optimization. timization problem. However, with a change of variables:
This work was supported by the Hertz Foundation Fellowship and the Stan- yi = log xi and bik = log cik , it can be shown that the re-
ford Graduate Fellowship. formulated problem is a convex optimization problem [1].
III. C ONVEX O PTIMIZATIONS OF S INGLE Q UEUES B. Optimizing M/M/m/m queues
A. Optimizing for average delay and queue occupancy We now optimize specific queue occupancy probabilities by
first considering an M/M/m/m queue. The steady state
We start the suite of convex optimization formulations with a ( µ )k k!
λ 1
simple example of minimizing the service load of a M/M/1 probability of state k is given by pk = m
( λ )i i!
1 . In many
i=0 µ
queue with constraints on average queuing delay W , total de- applications of queuing systems to network design, we would
lay D, and queue occupancy Q: like to maximize the probability of a particular desirable state,
Proposition 1: The following nonlinear optimization is a ge- without making the probabilities of other states too small. For
ometric program, and therefore can be turned into a convex example, we may want to design a telephone call service cen-
optimization and efficiently solved for its global optimum: ter so as to maximize the probability that a particular number
of telephone lines (e.g., 90%) are in use at any given time. We
minimize µ also want to jointly optimize the fairness parameters Cj that
λ
subject to W ≤ Wmax , bounds pj .
D ≤ Dmax , Proposition 2: The following nonlinear optimization of
(3) M/M/m/m queues is a geometric program:
Q ≤ Qmax ,
λ ≥ λmin ,
µ ≤ µmax maximize pk
subject to pj ≥ Cj , ∀j,
where the optimization variables are the arrival rate λ and the Cj ≥ Cj,min , ∀j, (5)
service rate µ. The constant parameters are the performance λ ≥ λmin ,
upper bounds Wmax , Dmax and Qmax , and practical con- µ ≤ µmax
straints on the maximum service rate µmax of the queue that
cannot be exceeded, and the minimum incoming traffic rate where the optimization variables are λ, µ and Cj , and the con-
λmin that must be supported. The objective is to minimize stant parameters are λmin , µmax and the fairness constraints
the service load. We can also show that even a joint opti- Cj,min , j = 1, 2, · · · , m.
mization over both (λ, µ) and (Wmax , Dmax , Qmax ) is still a This geometric programming formulation can be extended to
geometric program. maximize the probability for the state with the lowest prob-
The above formulation can be extended to a Markovian queu- ability to enforce maxmin fairness. Similar formulations can
ing system with N queues sharing a pool of service rate be done for parallel M/M/m/m queues and a general M/G
bounded by µmax (for example, connected to a common out- queue.
going link). The arrival rate to be supported for each indi-
vidual queue i is bounded by λi,min . There are delay and C. Optimizing M/M/1 queues
queue occupancy bounds Wi,max , Di,max and Qi,max for
each queue i. The objective now becomes minimizing a We now turn to M/M/1 queues, where the convexity property
weighted sum of the service loads for all the queues: is, surprisingly, more complicated than that of queues with fi-
Corollary 1: The following nonlinear optimization is a geo- nite buffer size. We first prove the convexity properties of
metric program: the relevant quantities and then show the appropriate nonlin-
ear optimization formulations. For M/M/1 queues, the state
N probability pk can be viewed as a function of either the traffic
minimize i=1 αi µii
λ
λ load ρ = µ or of λ and µ.
subject to Wi ≤ Wi,max ,
The state probability p1 is always a concave function of ρ.
Di ≤ Di,max ,
(4) However, there is an interesting transition from convexity to
Qi ≤ Qi,max ,
concavity as load increases for pk , k ≥ 2, derived from the
λi ≥ λi,min ,
N second derivative test of convexity and shown in the following
i=1 µi ≤ µmax
Lemma 1: The state probability pk , k ≥ 2 is a convex func-
where the optimization variables are the arrival rates λi and tion of load ρ if and only if (k − 1) − (k + 1)ρ ≥ 0.
the service rates µi . Therefore, there is a transition from convexity to concavity
A simple numerical example for N = 2 with weights across the states in ascending order as load increases from 1 3
α1 = 1, α2 = 2 is summarized as follows. If we set to 1. In order for pk to be convex in ρ for all k greater than or
the delay and queue occupancy constraints as Q1,max = equal to a critical k0 , the traffic load ρ must be smaller than a
4, Q2,max = 5, W1,max = 2.5, W2,max = 3, D1,max = critical ρ0 (k0 ). Numerically evaluating the above condition,
2, D2,max = 2, and service and arrival rate constraints as we obtain the convexity transition curve shown in Figure 1,
λ1,min = 0.5, λ2,min = 0.8, µmax = 3, geometric program- where the critical load ρ0 can be read for any given k0 .
ming gives the optimizers: µ∗ = 1.328, µ∗ = 1.672, λ∗ =
1 2 1 We now turn to the more useful design problem where we can
0.828, λ∗ = 1.172 and the optimized objective value is 4.457.
2 vary the arrival rate and service rate independently, instead of
B−1
Since pB = 1 − i=0 pi , this heuristics essentially min-
Critical load for convexity transition of state probabilities
1
0.9
imizes the blocking probability. It is also known that due to
the superadditive effect of buffer size on pB , allocating a fixed
0.8
0.7
0.6
buffer space among several queues to minimize the overall
critical rho
0.5 blocking probability is also a convex optimization.
0.4
An alternative way to characterize buffer overflow is through
0.3
the large deviation approach, where the blocking probability is
guaranteed statistically: for a connection X with a prescribed
0.2
0.1
0
5 10 15 20 25 30 35 40 45 50
service rate R in the queue, we would like to ensure that the
critical k
probability of overflow (receiving more than R bps from X)
Fig. 1. Threshold loads for transition of pk (ρ) from convexity to concavity over a time scale of t is exponentially small:
for a M/M/1 queue.
t
Prob X(i) ≥ R ≤ exp(−sR) (7)
i=1
just their ratio ρ. Unlike M/M/m/m queues where geomet-
ric programming can be used for optimizing over λ and µ, the where s ≥ 0 is the undersubscription factor. Smaller s im-
state probabilities pk of an M/M/1 queues are not in general plies more aggressive statistical multiplexing of multiple con-
convex functions of λ and µ. There is a similar, though more nections to one queue. This number R is called the effective
complicated, pattern of this transition when pk is viewed as bandwidth EB of X (as first proposed in [5], used in many
a function of two variables λ and µ. Interestingly enough, papers since, and nicely reviewed in [7]).
this transition pattern still only depends on ρ, as shown in the Using the Chernoff bound, the effective bandwidth is given by
following
Lemma 2: The function pk , k ≥ 2, is convex in λ and µ if and 1
EB(X) = log E [exp(sX)] , (8)
only if (k 2 +k)−(k 2 +k)ρ+(k 2 −k)ρ2 −(k 2 +k +2)ρ3 ≥ 0. st
This lemma leads to the following
and in practice, the expectation is replaced by empirical data
Proposition 3: The following nonlinear optimization of ˜
collected over a time period of t that is much larger than the
M/M/1 queues is a convex optimization problem: time scale factor t:
t
minimize pk ˜
t
pj ≤ Cj , j > k, 1 t
subject to EB(X) = log exp(sX(i))
µ ≤ µmax , (6) st ˜
t i=1
λ ≥ λmin ,
ρ < ρ0 (k) where X(i) is the number of bits produced by connection X
during the ith time slot.
where 0 ≤ ρ0 (k) < 1 solves the equation (k 2 + k) − (k 2 +
We want to either minimize the assigned service rate EB(X)
k)ρ + (k 2 − k)ρ2 − (k 2 + k + 2)ρ3 = 0. The optimiza-
subject to constraints that lower bound the traffic intensity
tion variables are λ and µ, and the constant parameters are
X(i) to be supported (i.e., exponentially small probability of
Cj , λmin , µmax and k.
overflow or blocking), or maximize the traffic intensity sub-
ject to constraints upper bounding the service rate that can be
IV. O PTIMIZATIONS W ITH B UFFER OVERFLOW assigned to X. Both problems are geometric programs, and
C ONSTRAINTS T HROUGH E FFECTIVE BANDWIDTH we focus on the first formulation for the rest of this section
(since it is also connected with information theoretic channel
One approach to study the buffer overflow probability is capacity [3]), where we put various constraints (indexed by j
through the blocking probability of an M/M/1/B queue with and induced by the stochasticity of other connections sharing
a fixed buffer of size B: the queue buffer) on the minimal level of traffic intensity to be
λ 1
( µ )B B! supported by EB(X).
pB = B λ i1
. Proposition 4: The following problem of constrained buffer
i=0 ( µ ) i! allocation through the effective bandwidth approach is a geo-
metric program:
Therefore, minimizing pB is equivalent to maximizing a
posynomial of λ and µ, which is in turn equivalent to max- minimize EB(X)
imizing a convex function. Therefore, it is not a convex opti- (9)
subject to i Pij X(i) ≥ Xmin,j , ∀j
mization problem. One possible heuristics is to use geomet-
ric programming to maximize the probability of some state where the optimization variables are X(i), and the constant
k, k < B, subject to lower bounds on pj for all other j < B. parameters are Pij and Xmin,j .
An illustrative numerical example is summarized as follows. Proposition 5: The nonlinear problem of maximizing the
With s = 0.5, t = 5ms, we impose a set of 10 different con- probability of state (n1 = n∗ , · · · , nK = n∗ ) with
1 K
N
straints to specify the type of an arrival curve a queue should k=1 nk = N , subject to fairness constraints on other states,
be able to support without blocking. The geometric program- is a geometric program:
ming solution returns the minimized effective bandwidth as
EB∗ (X) = 1.7627Mbps, and the envelope of supportable ar- maximize p(n1 = n∗ , · · · , nK = n∗ )
1 K
rival curves is shown in Figure 2. Connections with arrival subject to p(n1 , · · · , nK ) ≥ Fairness constants,
curves below this envelope will not cause buffer overflow or µi ≤ µi,max , (10)
queue blocking with a probabilistic guarantee as in (7). µi ≥ µi,min ,
K
Arrival curve constraint envolope k=1 mk µk ≤ µtotal
3
where there is a constraint of the first type for each steady
2.5 state probability p(n1 , · · · , nK ). The optimization variables
are µk , and the constant parameters are µi,min , µi,max and
Number of bits allowed in each ms (kb)
2
µtotal .
1.5
The above convex optimization problem can be viewed as a
problem of resource (i.e., service capacity µk ) allocation in a
1 closed queuing network. The goal is to maximize the proba-
bility that the system is in a particular state subject to fairness
0.5
constraints on other states and the limited system resource.
At first glance, it may seem that the above formulations can
0
0 10 20 30 40
Time (ms)
50 60 70 80
be readily extended to an open queuing network. How-
ever, because of a more complicated convexity structure of
Fig. 2. The envelope of arrival curves supportable by EB∗ (X) = 1.7627.
an open network (in particular, the steady state probability
V. C ONVEX O PTIMIZATIONS OF Q UEUING N ETWORKS p(n1 , n2 , · · · , nK ) is neither concave nor convex in ρk ), a sim-
ilar formulation can be intractable for a general open network
In some queuing problems, a fixed number of customers or of queues. For a simple example, consider an open queu-
tasks circulate indefinitely in a closed network of queues. For ing network with two interconnected M/M/1 queues, each
example, some computer system models assume that at any with an exponential service rate µk and an external arrival
given time a fixed number of programs occupy the resource. rate αk , k = 1, 2. After being served by queue k, a customer
Such problems can be modelled by a closed queuing network chooses to go to the other queue with probability pk or leave
consisting of K nodes, where each node k consists of mk the queuing system with probability 1 − pk . It can be shown
identical exponential servers, each with average service rate that p(n1 , n2 ) is neither concave nor convex.
µk . There are always exactly N customers in the system.
Once served at node k, a customer goes to node j with prob- VI. C ONVEX O PTIMIZATIONS OF F EEDBACK C ONTROL
ability pkj . Then for each node k, the average arrival rate to IN Q UEUING N ETWORKS
K
the node, λk , is given by λk = j=1 pkj λj .
In this section, we extend the convex optimization formula-
The steady state probability that there are nk customers in
tions to a particular type of queuing networks with feedback.
node k, for k = 1, 2, · · · , K, is given by [8] (a closed network
Although the formulations are no longer in the special form
Jackson’s theorem):
of geometric program, we can still turn them into a general
nk convex optimization problem.
K λk
1 µk
As a first example, consider a simple network of queues
p(n1 , n2 , · · · , nK ) = ,
G(K) βk (nk ) shown in Figure 3.
k=1
where Incoming Queue 1
Packets
nk ! nk ≤ mk
βk (nk ) =
mk !mnk −mk
k nk > mk Queue 2
and the normalization constant G(K) is given by Fig. 3. A network of queues with feedback.
λk
nk The incoming traffic to the overall system is i.i.d. ∼
N
µk Poisson(λ). With probability p1 , an incoming packet leaves
G(K) = the system after the feedforward queue 1, which has an ex-
s
βk (nk )
k=1
ponential service time µ1 , and with probability p2 = 1 − p1 ,
where the summation is taken over all state vectors the packet is feed back through queue 2, which has an expo-
N
(n1 , n2 , · · · , nK ) satisfying k=1 nk = N . We have nential service time µ2 . This queuing model can be used for
solution for a fixed α. Note that only points to the right of the
a variety of systems where we would like to process as many
Pareto optimality curve are achievable.
packets through the feedback loop as allowed under a delay
constraint on the total time spent in the system. For example, 0.3
Pareto optimal tradeoff curve
in some optical packet switch architectures, the problem of
wavelength contention can be solved by cycling packets not 0.25
switched in a time slot through the buffer again. Clearly, there
is a tradeoff between maximizing the feedback queue traffic
1 0.2
load ρ2 (or equivalently, minimizing the service load ρ2 ) and
rho2
minimizing the total time T spent in the system. We have
0.15
Proposition 6: The nonlinear optimization problem of mini-
1
mizing both T and ρ2 by varying p2 , subject to the following
0.1
constraints: ρ1 < 1, ρ2 < 1, and 0 ≤ p2 ≤ 1 is a convex
multi-objective optimization problem, where all the Pareto
0.05
optimal solutions can be found through a scalarization tech- 0 0.5 1 1.5 2 2.5
T
3 3.5 4 4.5 5
nique as follows.
1 Fig. 5. Pareto optimality tradeoff curve as α varies.
The problem of minimizing a weighted sum of T and ρ2 , sub-
ject to stability constraints for each individual queue, is a con- With two parallel feedback queues, maximizing a weighted
vex optimization problem in variable p2 : sum of feedback queue loads subject to upper bounds on the
feedforward queue load is a convex optimization, so is mini-
1
minimize T + α ρ2 mizing the ratio of the feedforward load and the sum of feed-
subject to ρ1 < 1, back loads. However, due to more involved convexity struc-
ρ2 < 1, (11) tures of the queuing system, extending the above analysis to a
p2 ≤ 1, general case with n feedback queues is not straightforward,
p2 ≥ 0
Therefore, the nonlinear problem of finding the best feedback VII. C ONCLUSIONS
parameter p∗ and p∗ = 1 − p∗ to minimize the total system
2 1 2 Based on various results on convexity properties of queuing
time and maximize the feedback queue traffic load under indi-
systems and computationally efficient algorithms for convex
vidual queue stability constraints can be efficiently solved for
optimization, we present a suite of formulations to optimize
the globally optimal solution.
the performance of single queues, networks of queues, and
We summarize a numerical example for the case of one feed- large deviation theoretic bounds on blocking probability min-
back queue. Starting with the scalarized version, we first fix a imization. These nonlinear performance optimizations can be
weighting factor α = 0.5 for λ = 5, µ1 = 8 and µ2 = 8. As carried out globally in polynomial time for network queuing
shown in Figure 4, the convex optimization algorithm finds
systems under QoS and fairness constraints.
the global optimum value for the objective function as 3.933
through the optimal feedback probability p∗ = 0.2755.
2
R EFERENCES
Minimization of the scalarized problem for alpha=0.5
100
[1] S. Boyd and L. Vandenberghe, Convex Optimization Stanford Univer-
90 sity EE 364 Course Reader 2001.
80
[2] C. S. Chang, “Stability, queue length and delay of deterministic and
stochastic queuing networks,” IEEE Trans. Automatic Control, vol. 39,
70
pp. 913-931, 1994.
Objective function value
60 [3] M. Chiang and S. Boyd, “Shannon duality through Lagrange duality:
efficient computation and free energy interpretations for channel ca-
50
pacity and rate distortion,” Proc. 40th Allerton Conference, Oct. 2002.
40 [4] R. J. Duffin, E. L. Peterson, C. Zener, Geometric Programming: The-
30
ory and Applications, Wiley 1967.
[5] J. Y. Hui, “Resource allocation for broadband networks,” IEEE J. Sel.
20
Area Comm., pp.1598-1608, 1988.
10 [6] D. Julian, M. Chiang, D. O’Neill, and S. Boyd, “QoS and fairness con-
0
strained convex optimization of resource allocation in wireless cellular
0 0.05 0.1 0.15 0.2
p2
0.25 0.3 0.35 0.4
and ad hoc networks,” Proc. IEEE Infocom, New York, June 2002.
[7] F. P. Kelly, “Notes on effective bandwidth,” Stochastic Networks: The-
α ory and Applications, pp.141-168, Oxford Unviersity Press, 1996.
Fig. 4. Optimizing T + ρ2
over p2 for a fixed α.
[8] L. Kleinrock, Queueing Systems, vol.I, Wiley, 1974.
Now we solve the multi-objective problem as in Proposition 6 [9] Yu. Nesterov and A. Nemirovsky, Interior Point Polynomial Method
in Convex Programming, SIAM 1994.
through scalarization of this convex Pareto optimization. Due
to the convexity structure, by varying α we can obtain the en-
tire tradeoff curve shown in Figure 5, where each point on the
curve corresponds to the result of a convex scalar optimization
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