Queueing Syst (2011) 68:217–220
DOI 10.1007/s11134-011-9259-5
Introduction to special issue on open problems
Published online: 11 August 2011
© Springer Science+Business Media, LLC 2011
This special issue contains a set of open problems in the area of applied probability,
and, more specifically, (stochastic) queueing systems. The primary idea behind this
initiative is that we strongly feel that it could serve as an eye-opener for young researchers who want to get a feel for the open research questions. In addition, a list of
clearly stated open problems could also challenge more experienced specialists. The
open problems thus provide the readership with a state of the art, and will hopefully
trigger new efforts, new collaborations, and new approaches.
All papers given in this issue include a clear, self-contained description of the open
problem (or problems), the discussion of its (their) significance and the current state
of understanding for this problem (these problems), including known partial solutions
and an account of the most relevant literature related to the subject. In addition, if the
open problem is in the form of a conjecture, there is some motivation why the author/authors thinks/think the conjecture is true. To some papers a short discussion of
a referee is attached proposing new approaches to solve the problem and/or directions
for further research.
A website, containing new developments regarding the open problems, will be
maintained by James Cruise at http://www.ma.hw.ac.uk/~rc141/QUESTA/.
During the meeting ‘Stochastic Networks and Related Topics’ in Bȩdlewo, Poland,
in May 2009, Sergey Foss invited us to lead this special issue. As we hardly had a realistic impression of the amount of work it would take us, we accepted. . . . Now, about
two years later, we feel this volume contains nice accounts of the most important
open problems in our area. The programme ‘Stochastic Processes in Communication
Sciences’ at the Isaac Newton Institute in Cambridge, UK, as well as the workshop
‘Mathematical Challenges in Stochastic Networks’ in Oberwolfach, Germany, also
played an important role in the preparations of this special issue.
The nineteen papers in this issue cover a variety of challenging topics ranging from
(A) stability, through (B) tail asymptotics, (C) the theory of Markov processes, and
(D) stochastic geometry, up to (E) classical queues, polling systems and scheduling.
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Queueing Syst (2011) 68:217–220
(A) Stability. Rojas-Nandayapa, Foss, and Kroese consider a queueing system in
which arriving customers are placed on a circle and wait for service. A traveling
server moves at constant speed on the circle, stopping at the location of the customers
until service completion. The server is greedy: always moving in the direction of the
nearest customer. It was earlier conjectured that this system is stable if the traffic
intensity is smaller than 1; this review presents an account of the current state of this
conjecture, and suggests various related open problems.
Antunes, Fricker, and Roberts consider a multi-server polling system where the
number of servers that can attend to a queue simultaneously is limited. Stability conditions are available for the case of unlimited and limited service policies when the
server limits values are equal to the total number of servers. The case of general
server limits and limited service appears to be much harder; a conjecture is stated for
the stability condition in this case, supported by simulation-based evidence.
Walton and Mandjes describe the current state of affairs in the area of stability for
α-fair networks. Several special cases have been rigorously dealt with (exponentially
distributed jobs, phase-type jobs, special topologies, specific values of α, etc.), but in
its full generality the problem has not been solved so far. Thomas Bonald’s review
helped improving the presentation of the results.
(B) Tail asymptotics. The paper of Miyazawa and Kobayashi gives conjectures
regarding the tail decay of the stationary distribution of a d-dimensional semimartingale reflecting Brownian motion (SRBM).
Then Korshunov discusses the problem of establishing an upper bound for the
distribution tail of the stationary waiting time D in the GI/GI/1 FCFS queue. It is
known that if the residual service time Br is subexponential distribution, then
P(D > x) ∼
ρ
B r (x),
1−ρ
where ρ < 1 is a traffic load. The paper assesses how accurate such an approximation
is.
Dȩbicki and Mandjes present three challenging open problems that originate from
the analysis of the asymptotic behavior of Gaussian fluid queueing models. In particular, the problem of characterizing the correlation structure of the stationary buffer
content process is addressed. Moreover, the authors consider the speed of convergence to stationarity, and the analysis of an asymptotic constant associated with the
stationary buffer content distribution.
Then Kulik and Palmowski are interested in the areas under the workload graph, as
well as the area under queueing graph, during a busy period, focusing on the single
server queue. Their primary interest is in the asymptotic tail of the respective distributions. Both light-tailed and heavy-tailed service times are discussed. It is pointed
out how these results can be of importance also in percolation theory and actuarial
science. Their contribution benefited significantly from remarks by Ken Duffy and
Sean Meyn.
Duffy and Meyn study Loynes’ distribution, which characterizes the one-dimensional marginal of the stationary solution to Lindley’s recursion, for the case it has
an ultimately exponential tail. The open problems relate to estimating techniques for
the tail exponent of Loynes’ distribution. It is conjectured that under quite general
Queueing Syst (2011) 68:217–220
219
conditions a consistent sequence of non-parametric estimators can be constructed
that satisfies a large deviation principle.
(C) Markov process theory. Dai and Dieker describe an open problem for two
classes of d-dimensional diffusion processes: semimartingale reflecting Brownian
motions (SRBMs) and piecewise Ornstein–Uhlenbeck (OU) processes. The open
problem concerns the question when a solution to a basic adjoint relationship (BAR)
associated with such a diffusion process does not change sign.
Glynn’s open problem concerns Harris recurrence, which is a widely used tool in
the analysis of queueing systems. While such results are available for discrete-time
Harris chains, in continuous time the question of whether all Harris recurrent Markov
processes are automatically wide-sense regenerative is an open problem.
(D) Stochastic geometry. Thorisson presents four sets of related open problems in
renewal, coupling and Palm theory.
Hellings, Borst, and van Leeuwaarden introduce a class of tandem queueing networks which arise in modeling the congestion behavior of wireless multi-hop networks with distributed medium access control. These models, falling at the interface
between classical queueing networks and interacting particle systems, provide valuable insight in how the network performance in terms of throughput depends on the
back-off mechanism that governs the competition among neighboring nodes for access to the medium. Various open problems and conjectures are presented, backed by
partial results for special cases and limit regimes as well as simulation experiments.
Bordenave, Foss, and Last introduce a greedy walk on Poisson and Binomial processes which is a close relative to the well known greedy server model (as was described above in the context of the contribution of Rojas-Nandayapa et al.). A number
of open problems are presented.
(E) Classical queues, polling systems, and scheduling. First Gupta and Osogami
postulate a conjecture regarding upper and lower bounds on the mean waiting time
for the M/G/K queue, analogously to the classical Markov–Krein theorem. Suggested
by the proof for the case of n = 3 under light-traffic conditions, the authors believe
that a more promising approach would be to start with the Lindley recursion for the
Kiefer–Wolfowitz workload vector, which could also be useful for other queueing
problems represented by the fixed point of a stochastic recursive equation. The authors provide numerical evidence that support the conjecture using the M/H2 /K and
M/H3 /K queues with K = 4, 5.
Veeger, Kerner, Etman, and Adan study the standard M/G/s queue. With the random variable Dn denoting the time that elapses from a departure leaving behind n
customers in the system until the next departure, the authors are interested in the
limit of E(Dn ) as n tends to infinity (if it exists) as well as distributional properties
of the sequence Dn .
Janssen, van Leeuwaarden, and Zwart deal with the classical Erlang C formula for
the M/M/s queue describing the probability C(s, a) that a customer is delayed when
arriving at the system in steady
√ state, where a is the offered load. Halfin and Whitt
showed that setting s = a + β a (square-root staffing) for any fixed β > 0 yields
√
Φ(β) −1
,
lim C a + β a, a = C∗ (β) := 1 + β
a→∞
φ(β)
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Queueing Syst (2011) 68:217–220
with Φ, φ the distribution and density of the standard normal random variable. The
authors describe other properties of the above convergence, and postulate conjectures.
This contribution has benefited from useful remarks by David Goldberg.
Boon, Boxma, and Winands describe two open problems concerning polling systems. The first open problem deals with a system consisting of two queues, one of
which has gated service, while the other receives 1-limited service. The second open
problem concerns polling systems with general (renewal) arrivals and deterministic
switch-over times that become infinitely large.
Shah, Tsitsiklis, and Zhong review some known results and state a few versions of
an open problem related to the scaling of the total queue size (in steady state) in an
n × n input-queued switch, as a function of the port number n and the load factor ρ.
The paper concerns the question whether the total number of packets in queue, under
either the maximum weight policy or under an optimal policy, scales (ignoring any
logarithmic factors) as O(n/(1 − ρ)).
Nazarathy considers the variance of the departure process from a queue. He looks
in particular at the BRAVO (Balancing Reduces Asymptotic Variance of Outputs) effect known to occur in M/M/1/K and GI/G/1 queues, with factors 1/3 asymptotically
and 1 − 2/π , respectively. Some numerical and simulation work is carried out that
supports the hypothesis that the same effect is present in the GI/G/1/K queue. A hypothesis on the value of the factor is also presented. This paper has used valuable
insights from the review by Daryl Daley; we have decided to include his report as a
separate paper.
We like to thank the authors for their excellent contributions as well as the referees
for their efforts. It has been a privilege for us to serve as guest editors and work with
Sergey Foss, and the editorial manager Aiza Mae F. Policarpio of Springer. We are
grateful for their valuable advice, guidance and support.
May 2011
Amsterdam
Wrocław
Edinburgh
M. Mandjes
Z. Palmowski
V. Shneer