arXiv:2505.00198v1  [math.PR]  30 Apr 2025
Queueing models with random resetting
GUODONG PANG∗, IZABELLA STUHL†, AND YURI SUHOV$
Abstract. We introduce and study some queueing models with random resetting, includ-
ing Markovian and non–Markovian models. The Markovian models include M/M/1, M/M/r
and M/M/∞queues with random resetting, in which a continuous-time Markov chain is
formulated and the transition from each state includes a resetting to state zero in addition
to the arrival and service transitions. Hence the chains are no longer a birth and death
process as in the classical models. We explicitly characterize the stationary distributions of
the queueing processes in these models. It is worth noting the distinction of the stability
conditions from the standard models, that is, the positive recurrence of the Markov chains
does not require the usual traﬃc intensity to be less than one.
The non–Markovian models include GI/GI/1, GI/GI/r and GI/GI/∞queues with ran-
dom resetting to state zero. For GI/GI/1 and GI/GI/r queues, we consider random resetting
at arrival times, and introduce modiﬁed Lindley recursions and Kiefer–Wolfowitz recursions,
respectively.
Using an operator representation for these recursions, we characterize the
stationary distributions via convergent series, as solutions to the modiﬁed Wiener–Hopf
equations. For GI/GI/1 queues with random resetting, a particularly interesting case is
when the diﬀerence of the service and interarrival times is positive, for which an explicit
characterization of the stationary distribution of the delay/waiting time is provided via the
associated characteristic functions. For GI/GI/∞queues, we also consider random reset-
tings at arrival times, by utilizing a version of the Kiefer–Wolfowitz recursion motivated from
that for GI/GI/r queues, and also characterize the corresponding stationary distribution.
1. Introduction
In this paper we study some queueing models with random resetting, in which a queue
clears when resettings occurs. Such models have many applications in service systems where
machines or servers are subject to maintenance after some random time or periodically, or
where disruptions occur due to power loss or breaking down. There exists a substantial liter-
ature in queueing theory to model such phenomena, such as queues in random environments,
queues with disasters and so on. In this work we aim at developing a number of queue-
ing models extending the standard ones in a uniﬁed manner to take into account random
resettings and treating both Markovian and non–Markovian models.
We start with the Markovian models with random resetting, extending the standard
M/M/1, M/M/r and M/M/∞systems.
The queueing process in these classical models
is a birth–death process with jumps ±1, whose stationary distribution is known explicitly.
Date: May 2, 2025.
Key words and phrases. Queues, random resetting, stationary distribution, M/M/1, M/M/∞, M/M/r,
GI/GI/1, GI/GI/r, GI/GI/∞, modiﬁed Lindley recursion with random resetting, modiﬁed Kiefer–Wolfowitz
recursion with random resetting, modiﬁed Wiener–Hopf equation, resetting at arrival times.
1

2
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
For M/M/1 and M/M/r queues, this leads to a stability condition: the traﬃc intensity per
server is less than one. In a system with resetting, the queueing process is a continuous-
time Markov chain (CTMC) which not only can jump by ±1, but also can hop to state 0;
consequently, the CTMC is no longer a birth–death process. Here the transitions are charac-
terized by arrival, departure and resetting rates. It turns out that for the above models, the
CTMC with resetting is always positive recurrent (even if the departure rate equals 0), and
its stationary distribution can be written down explicitly. See Theorems 3.1, 2.1, and 4.1.
Similar results have been obtained for stochastic clearing models of point processes studied
in [24, 23, 28].
Next, we pass to non–Markovian models with random resetting, extending the standard
GI/GI/1 and GI/GI/r systems under the FCFS discipline and the GI/GI/∞system. For a
standard GI/GI/1 queue, the Lindley recursion is fundamental in studying the delay/waiting-
time behavior (see Section 5.1 for a brief review and [2, Chapter X.1]). Similarly, for a stan-
dard GI/GI/k queue, it is the Kiefer–Wolfowitz recursion that determines the delay/waiting–
time behavior (see Section 6.1). We study the corresponding models with random resetting
at customers’ arrival times, which leads to modiﬁed Lindley and Kiefer–Wolfowitz recursions
(see equations (5.4) and (6.7)).
For standard GI/GI/1 and GI/GI/r queues, the positive recurrence of the waiting–time
process requires that the traﬃc intensity per server is less than one, and then the stationary
distribution is characterized via the Wiener–Hopf equation. For the same models with ran-
dom resetting, we show that the corresponding modiﬁed Lindley and the Kiefer–Wolfowitz
recursions always generate positive recurrent waiting–time processes, regardless of whether
their standard counterparts are positive recurrent or not. More importantly, the modiﬁed
Lindley and the Kiefer–Wolfowitz recursions can be conveniently represented as an operator
form (see equations (5.8) and (6.5)), from which we are able to express the stationary dis-
tribution as a convergent series (see equations (5.10) and (6.9)-(iv)). As a byproduct, we
have an interesting ﬁnding for GI/GI/1 queues with random resetting when the diﬀerence
of the service and interarrival times takes positive values (unlike the standard case assuming
negative mean), that the stationary distribution for the waiting time/delay can be explicitly
expressed via the characteristic functions of the diﬀerence variable mentioned above, see
equation (5.14).
Finally, we consider GI/GI/∞models with random resetting at arrival times. We ﬁrst
construct a recursion for the elapsed service times for the jobs in service in GI/GI/∞queues,
by adapting the Kiefer–Wolfowitz recursion for GI/GI/r queues, and then use it to formulate
the recursion for the GI/GI/∞queues with resetting. We show that the recursion is positive
recurrent and also derive an explicit expression for the associated stationary distribution,
similarly as the GI/GI/r queues with resetting (see Section 7).
A review of the literature.
The models discussed in this paper are related to several
streams of the existing literature. First, these models are related to the stochastic clearing
models studied in [24, 23, 28], where a stochastic input process (such as the arrival process)
is intermittently and instantaneously cleared. Various clearing policies have been studied,
e.g., clearance when the input reaches a threshold, or at i.i.d. random times independent

3
of the input process. Our Markovian models without service can be regarded as stochastic
clearing models of Poisson arrivals at exponentially distributed random times, while our
non–Markovian models without services can be regarded as stochastic clearing models of
renewal arrivals at arrival times. However, the stochastic clearing models do not have output
dynamics like our models.
Second, our models are related to the queueing models with disasters, see, e.g, [6, 10, 16,
3, 11, 19, 29, 7, 26]. For example, in [6], an M/G/1 queue with “disasters” has been consid-
ered, where disasters occur at certain random times including (a) deterministic equidistant
times, (b) random times independent of the queueing process, and (c) at crossings of some
pre-speciﬁed level. In these works, stationary distributions of the workload processes have
been characterized via their Laplace transforms using certain modiﬁcations of the Lindley
recursion.
For another example, the paper [16] considers an M/M/1 queue with catastrophes, where
the server breaks downs at i.i.d. random times, independent of the service process. At the
breakdown times, all jobs are lost, and it takes an exponential random time to repair the
system. See also similar formulations of “catastrophes” or “clearing” in [10, 3, 11, 19, 26].
An associated (jump) diﬀusion approximation has been considered in [10, 9, 8]. In [13], a
computational approach has been developed, for non-homogeneous Markovian single-server
and inﬁnite-server queueing models, whose formulation is like our Markovian models but
with nonstationary transition rates. A more general birth-death process with catastrophes
is studied in [9].
Our study clearly distinguishes from the existing literature of queues with disasters since
the Markovian models have explicit stationary distributions, and the non–Markovian models
with random resetting at arrival times are completely new. We also refer the readers to some
recent studies of random walks, Brownian motions and diﬀusions with random resetting in
[12, 25, 27, 1, 18].
Unlike the “clearing” phenomenon in the models described above, there are some recent
studies of queues with random resetting in [4, 22], where the authors study a single server
M/G/1 queue with service times being reset at random times whenever the service time is
longer than a threshold. This concept of stochastic resetting is also exploited in random
search problems, Cf. [5, 20].
Organization of the paper. The paper is organized as follows. The Markovian models with
resetting are studied ﬁrst, with M/M/∞queues in Section 2, M/M/1 queues considered in
Section 3, and M/M/r queues in Section 4. The non–Markovian models are studied next,
with GI/GI/1 and GI/GI/r queues with resettings at arrival times in Sections 5 and 6,
respectively, and with inﬁnite-server queues with resettings at arrival times in Section 7.
2. The M/M/∞queue with random resetting
The standard assumption in Sections 2–4 is that the jobs arrive in a Poisson process at
rate λ > 0, the services times are i.i.d. exponential of rate µ ≥0, and all jobs in the system
are cleared/reset after after subsequent i.i.d. exponential random times of rate κ > 0. As

4
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
usual, we assume mutual independence of all the processes involved. The case of µ = 0
means that no jobs are served; in this case we get a stochastic clearing model of a Poisson
process, Cf. [24, 28].
This section focuses on an M/M/∞queue with resetting. Denote by X(t) the number of
jobs in the system at time t ≥0. Then {X(t) : t ≥0} is the continuous-time Markov chain
(CTMC) on Z+ = {0, 1, 2, . . .} with the transition rates
i ≥0
→
i + 1
rate
λ
(arrival),
i ≥1
→
i −1
rate
iµ
(departure),
i ≥1
→
0
rate
κ
(resetting).
(2.1)
The process {X(t)} with µ > 0 is dominated by the process with µ = 0. Alternatively,
{X(t)} is dominated by the standard M/M/∞queuing process, with the same µ and κ = 0.
We denote by π = {πi : i ∈Z+} the stationary distribution of process {X(t)}.
Theorem 2.1. Assume that µ > 0. The stationary distribution π for the M/M/∞queue
with resetting is given by
π0 = 1
λ

µ + κ
ρ(eρ −1)
µ
λ + κ/λ + 1
ρ
(eρ −1)
−1
,
πi =
µ
λ + κ/λ + 1
ρ
(eρ −1)
−1ρi−1
i!
,
i ≥1 .
(2.2)
Here ρ > 0 solves the equation:
κ(eρ −1) = −µρ2 + (λ + κ)ρ.
(2.3)
Proof. To characterize the stationary distribution, we have the partial balance equations
(PBEs):
λπ0 = µπ1 + κ
X
j≥1
πj ,
(λ + iµ + κ)πi = λπi−1 + (i + 1)µπi+1 ,
i ≥1 .
We use the following Ansatz:
πi = π1
ρi−1
i! ,
fori ≥1,
where ρ is to be determined.
By the above equation for π0, we have
λπ0 = µπ1 + κπ1
X
j≥1
ρj−1
j!
= µπ1 + κ
ρπ1(eρ −1) ,
whence
π0 = 1
λ

µ + κ
ρ(eρ −1)

π1 .
(2.4)
By the Ansatz, π2 = π1ρ/2; we substitute it into the PBE for π1. Together with the form of
π0, it leads to equation (2.3) which clearly has one positive solution.

5
Finally, equation (2.4) and the condition P
i≥0
πi = 1 provide the explicit expression for π1:
π1 =
µ
λ + κ/λ + 1
ρ
(eρ −1)
−1
.
(2.5)
Plugging this into (2.4), we obtain the claimed form of π.
□
Remarks. 2.1. The distribution π in (2.2) is a mixture of a Poisson distribution and the
Dirac delta at 0:
πi = αρie−ρ
i!
+ (1 −α)δi,0,
i ≥0,
(2.6)
where
α = eρ
ρ
µ
λ + κ/λ + 1
ρ
(eρ −1)
−1
∈(0, 1).
(2.7)
When κ = 0, that is, the standard M/M/∞queue, equation (2.3) reduces to ρ = λ/µ,
which implies π1 = ρe−ρ, and then π0 = ρ−1π1 = e−ρ and πi = e−ρρi/i! for i ≥1. That is, π
follows the Poisson distribution of parameter ρ.
2.2. For µ = 0 (the stochastic clearing model of Poisson process), the PBEs become
λπ0 = κ
X
j≥1
πj,
(λ + κ)πi = λπi−1 ,
i ≥1 .
(2.8)
It gives
πi = ̺i(1 −̺), i ≥0
(geometric), where
̺ :=
λ
λ + κ.
(2.9)
3. The M/M/1 queue with random resetting
In this section we consider an M/M/1 queue under the ﬁrst-come ﬁrst-served (FCFS)
discipline with random resetting. Let X(t) be the number of jobs in the system at time t.
The CTMC {X(t) : t ≥0} has the transition rates
i ≥0
→
i + 1
rate
λ
(arrival),
i ≥1
→
i −1
rate
µ
(departure),
i ≥1
→
0
rate
κ
(resetting).
(3.1)
When µ = 0, the CTMC {X(t)} is positive recurrent ∀λ, κ > 0: see (2.8). Owing to the
dominance, we obtain the following property.
Proposition 3.1. For any µ ≥0 and λ, κ > 0, the CTMC {X(t) : t ≥0}
is positive
recurrent and has a unique stationary distribution.
As before, let π = {πi : i ∈Z+} be the stationary distribution, which is also equal to the
limiting probability: πi = lim
t→∞P(X(t) = i) for i ∈Z+.

6
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
Theorem 3.1. Assume that µ > 0. The stationary distribution π is given by
π0 =
1
1 −β

1
1 −β +
1
1 −ρ
−1
,
πi = ρi−1
1
1 −β +
1
1 −ρ
−1
,
i ≥1 .
(3.2)
Here β ∈(0, 1) and ρ ∈(0, 1) are given by
β = 1 −
µ
λ + κ
λ
1
1 −ρ
−1
,
(3.3)
and
ρ = λ + µ + κ
2µ
−
s
(λ + µ + κ)2
4µ2
−λ
µ .
(3.4)
Proof. We know that π must satisfy the partial balance equations (PBEs) and have P
i
πi = 1.
It yields
λπ0 = µπ1 + κ P
j≥1
πj ,
(λ + µ + κ)πi = λπi−1 + µπi+1 , i ≥1 .
The Ansatz is now that πi = π1ρi−1 for some ρ ∈(0, 1) and i ≥1. Plugging it into the
above equation for π0, we obtain
λπ0 = µπ1 + κπ1
1
1 −ρ = π1

µ +
κ
1 −ρ

,
that is,
π0 = π1
µ
λ + κ
λ
1
1 −ρ

= π1(1 −β)−1 .
The equation for πi with i ≥2 yields
(λ + µ + κ)ρ = λ + µρ2.
From this equation, if µ = 0, then ρ =
λ
λ + κ (cf. (2.9)), and if µ > 0, then we have
ρ = λ + µ + κ
2µ
−
s
(λ + µ + κ)2
4µ2
−λ
µ ,
which can be easily checked to be in (0, 1). The other solution to the quadratic equation is
disregarded since it is bigger than 1.
Next, by P
i
πi = 1, we have
π0 + π1
1
1 −ρ = π1

1
1 −β +
1
1 −ρ

= 1 ,
which gives
π1 =

1
1 −β +
1
1 −ρ
−1
.
From here we obtain the expressions (2.1) and (2.2) as claimed.

7
Note that
1
1 −β = µ
λ + κ
λ
1
1 −ρ > 0 ,
which implies that β ∈(0, 1).
□
Remarks. 3.1. The distribution π in (3.2) is a mixture of a geometric distribution and
the Dirac delta at 0:
πi = αρi(1 −ρ) + (1 −α)δi,0,
i ≥0.
(3.5)
Here, α =
1 −β
ρ(1 −ρ + 1 −β) ∈(0, 1).
3.2. When µ = 0, we again obtain πi in the form of (2.9).
4. The M/M/r queue with random resetting
We now pass to an M/M/r queue under the FCFS discipline with random resetting. Let-
ting X(t) is the number of jobs in the system at time t, then {X(t) : t ≥0} is again a CTMC
on Z+. Here the transition rates are:
i ≥0
→
i + 1
rate
λ
(arrival),
i ≥1
→
i −1
rate
(i ∧r)µ
(departure),
i ≥1
→
0
rate
κ
(resetting).
(4.1)
It is evident that the CTMC {X(t)} has a unique stationary distribution π = {πi : i ∈Z+}.
Theorem 4.1. Assume that µ > 0. The stationary distribution π for the M/M/r queue with
random resetting is given by
π0 =
A/λ
A/λ + B ,
πi = (A/λ + B)−1ρi−1
i! ,
1 ≤i < r,
πi = (A/λ + B)−1 ρi−1
r!ri−r ,
i ≥r.
(4.2)
Here ρ ∈(0, r) is given by the following:
ρ = (λ + rµ + κ) −
p
(λ + rµ + κ)2 −4µλr
2rµ
,
(4.3)
whereas
A = µ + κB,
B =
r−1
X
i=1
ρi−1
i!
+
∞
X
i=r
ρi−1
r!ri−r .
(4.4)

8
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
Proof. To characterize the stationary distribution, we again use the PBEs:
λπ0 = µπ1 + κ
X
j≥1
πj,
(λ + iµ + κ)πi = λπi−1 + (i + 1)µπi+1,
1 ≤i < r,
(λ + rµ + κ)πi = λπi−1 + rµπi+1,
i ≥r.
(4.5)
We now use the Ansatz in the following form:
πi = π1
ρi−1
i! , 1 ≤i < r,
and
πi = π1
ρi−1
r!ri−r , i ≥r.
(4.6)
where ρ ∈(0, r) is a constant to be determined. From the top equation in (4.5), we obtain
λπ0 = Aπ1 .
(4.7)
Then, from the equation for πr, we get
(λ + rµ + κ)ρ = λr + rµρ2,
which yields ρ as in (4.3) (discarding the root > 1).
Next, the condition P
i≥0
πi = 1 gives an explicit form of π1:
π1 = (A/λ + B)−1 .
(4.8)
Then the expressions for π0 and πi are obtained from (4.7) and (4.8) whereas πi for i ≥1
are calculated from (4.6).
□
Remarks. 4.1. The distribution π in (4.2) is a mixture of a stationary distribution for a
sub-critical M/M/r queue and the Dirac delta at 0:
πi = αbπi + (1 −α)δi,0,
i ≥0.
(4.9)
Here,
bπi = C−1ρi ×
(
1/i!, 0 ≤i < r,
1/(r!rr−i), i ≥r,
where C =
 r−1
X
i=0
ρi
i! + rr
r!
r
r −ρ
!−1
(4.10)
and α = B + 1/ρ
A/λ + B ∈(0, 1).
4.2. In the limit as r →∞, we have that B →(eρ −1)/ρ, and A →µ + κ(eρ −1)/ρ, so
that A/λ + B →µ/λ + (1 + κ/λ)(eρ −1)/ρ. Hence, we recover the result for the M/M/∞
queue with resetting.
4.3. When κ = 0 and λ < rµ, we get ρ = λ/(rµ) < 1. Then A = µ and B = C, where C
is as in (4.10).

9
5. The GI/GI/1 queues with random resetting at arrival times
In this section, we consider a GI/GI/1 queue under the FCFS discipline with random
resetting at arrival times, particularly, focusing on the waiting times (delays) of jobs in the
system. We will be using [2] as a main reference book; the original works containing related
results can be traced via comments and the bibliography in [2].
5.1. The Lindley recursion for GI/GI/1 queue. The key ingredient of the GI/GI/1 model is
a sequence of real random variables (RVs) Xn, n ≥0, where Xn = Vn −Un, Un is the nth
inter-arrival time and Vn the nth service time. It is assumed that the RVs Xn are IID, with
a common cumulative distribution function (CDF) FX. In all of Sections 5–7, we assume
that FX is a proper CDF on R. The latter signiﬁes that
lim
x→−∞FX(x) = 0, lim
x→∞FX(x) = 1,
i.e., that the RV Xn take ﬁnite values only.
Let Wn be the nth waiting time. The Lindley recursive equation (originated in [17]) states:
Wn+1 = (Wn + Xn)+,
n ≥0,
(5.1)
with some given RV W0 ≥0, assumed to be independent of {Xn}. Here and below, we set
Y + = 0 ∨Y . Then {Wn, n ≥0} is a discrete-time Markov chain (DTMC) on R+ = [0, ∞).
It is known (see, e.g., [2, Chapter X.1]) that if E[X] < 0, there exists a unique stationary
distribution of DTMC {Wn}. (In fact, for E[X] < 0, the DTMC {Wn} is Harris ergodic.)
The stationary distribution is characterized by a proper CDF FW on R+ determined as a
unique solution to the stationary Wiener–Hopf (WH) equation
FW(t) = (FX ∗FW)(t)1R+(t),
t ∈R.
(5.2)
Here and below, G1 ∗G2 means the convolution of CDFs Gi:
G1 ∗G2(t) =
Z
R
G1(t −y)dG2(y) = G2 ∗G1(t),
t ∈R.
When E[X] ≥0, (5.2) has no solution among proper CDFs (again, see [2, Chapter X.1]).
A stochastic version of equation (5.2) reads
W
d= (W + X)+ .
(5.3)
Here X and W are ‘generic’ RVs with CDFs FX and FW, respectively, independent of each
other, and
d= means “equality in distribution”.
5.2. The modiﬁed Lindley recursion for a GI/GI/1 queue with resetting. We consider a
GI/GI/1 model where random resettings occur independently at arrival times. That is, the
(n + 1)st reset waiting time W R
n+1 either continues as in (5.1) with probability q ∈(0, 1), or
is set to be 0 with probability 1 −q, independently of (Xk, W R
k ) with 0 ≤k ≤n. Recursively,
it can be expressed as follows:
W R
n+1 = Zn+1(W R
n + Xn)+,
n ≥0.
(5.4)

10
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
Here {Zn : n ≥1} is a sequence of IID Bernoulli RVs with probability P(Zn = 0) = q =
1 −P(Zn = 1), independent of {Xn}. Equivalently, we can write
W R
n+1 =
(
0,
with probability q,
(W R
n + Xn)+,
with probability 1 −q,
(5.5)
independently of (Xk, W R
k ) with 0 ≤k ≤n.
Equations (5.4) and (5.5) are referred to as a modiﬁed Lindley recursion with resetting.
The sequence {W R
n} forms a DTMC on R+. We show that it is Harris ergodic: this implies
that the DTMC {W R
n} has a unique stationary distribution, and the corresponding CDF,
denoted by FW R, is proper on R+ and has FW R(0) > 0.
Proposition 5.1. For any q ∈(0, 1) and a sequence of IID RVs {Xn}, the DTMC {W R
n}
is Harris ergodic.
Proof. Observe that the process is regenerative (possibly after the ﬁrst cycle in case the
system starts from W R
0 > 0), with the cycles (W R
1 , . . . , W R
T) where W R
1 = 0 and W R
T for
T being geometric of parameter q. That is, the state 0 forms a regeneration set. Hence,
the time for the chain to return to state 0 has a ﬁnite mean. Then, Harris ergodicity is
straightforward.
□
The stationary CDF FW R is identiﬁed as a solution to a stationary WH equation with
resetting
FW R(t) =
h
q + (1 −q)(FW R ∗FX)(t)
i
1R+(t),
t ∈R,
(5.6)
or to its stochastic analog
W R d= Z(W R + X)+ .
(5.7)
Here X and W R are ‘generic’ RVs with CDFs FX and FW R, respectively, and Z is a Bernoulli
RV with P(Z = 0) = q = 1−P(Z = 1). Furthermore, the RVs X, W R and Z are independent,
and, as before,
d= means “equality in distribution”.
5.3. The operator calculus for a GI/GI/1 queue with resetting. It is convenient to write
equation (5.6) in an operator form:
FW R = q1R+ + (1 −q)KFW R
or, equivalently,
 I −(1 −q)K

FW R = q1R+.
(5.8)
Here the operator K acts on a CDF H by the convolution
(KH)(t) = (H ∗FX)(t)1R+(t),
t ∈R,
(5.9)
and I stands for the unit map. 1 In other words, if Y is an RV with CDF H, then KH is the
CDF of (X + Y )+ where X and Y are taken to be independent.
1Operator K is linear in the sense that K
 nP
i=1
αiGi

=
nP
i=1
αiKGi. for any α1, . . . , αn ≥0 with
nP
i=1
αi = 1
and CDFs G1, . . ., Gn. In this paper we do not explore further properties of K, including a speciﬁcation of
the domain in a linear space and the issue of its boundedness/continuity/compactness of K. A similar view
will be adopted in Sects 6 and 7.

11
Equation (5.8) is solved by the series
FW R =
 I −(1 −q)K
−1(q1R+) = q P
j≥0
(1 −q)jKj1R+
= q
h
1R+ + (1 −q)FX1R+ + (1 −q)2(FX ∗(FX1R+))1R+
+(1 −q)3(FX ∗((FX ∗(FX1R+))1R+))1R+ + . . .
i
.
(5.10)
If the series on the RHS of (5.10) converges, say, point-wise, and the limit is a proper CDF
with support in R+, then we get a workable representation of FW R.
In fact, we have the point-wise bounds involving FX+, the CDF of RV X+ = X ∨0:
1R+ ≥FW R ≥q
h
1R+ + (1 −q)FX+ + (1 −q)2FX+ ∗FX+
+(1 −q)3FX+ ∗FX+ ∗FX+ + . . .
i
=
 I −(1 −q)K+−1(q1R+)
(5.11)
where K+ acts on a CDF H by
(K+H)(t) = (H ∗FX+)(t),
t ∈R.
(5.12)
This is a consequence of stochastic ordering where W R ≥0 and Z(W R+X)+ ≤so Z(W R+X+).
The upper bound in (5.11) makes sure that the series in both (5.10) and (5.11) converge
point-wise on R (the non-trivial part is convergence on R+).
For methodological reasons, we will connect the series in (5.10) and (5.11) in the following
Theorems 5.1 and 5.2.
Theorem 5.1. Fix q ∈(0, 1) and suppose that the series in (5.11) gives a proper CDF on
R+. Then the series in (5.10) determines a proper CDF satisfying (5.6). Furthermore, for
any q ∈(0, 1), the equation (5.6) has a unique bounded solution, and this solution is given
by the series in (5.10).
Proof. The assertions of series convergence and validity of equation (5.8) follow from the
construction and bounds in (5.11). It remains to check uniqueness. Let G1 : R →R and
G2 : R →R be two bounded functions such that Gi(t) =
h
q + (1 −q)(Gi ∗FX)(t)
i
1(t ≥0),
for t ∈R, i = 1, 2. Set: G := G1 −G2, then
G(t) = (1 −q)(G ∗FX)(t)1R+(t),
t ∈R.
It implies that for the value γ := sup
t∈R
|G(t)|, we get γ ∈(0, ∞) and
γ ≤(1 −q)γ
or, iterating ,
γ ≤(1 −q)k,
∀
k ≥1.
As k →∞, it yields γ = 0.
□
We now turn to the case where RV X ≥0: FW R is determined by the series in (5.11), that
is,
FW R = q
h
1R+ + (1 −q)FX + (1 −q)2FX ∗FX + (1 −q)3FX ∗FX ∗FX + . . .
i
.
(5.13)

12
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
Let φ(θ) = E[eiθX] and ψ(θ) = E[eiθW R] be the characteristic functions (CFs) of X and
W R, respectively, for θ ∈R. Then (5.13) is equivalent to
ψ(θ) = q P
k≥0
(1 −q)k(φ(θ))k =
q
1 −(1 −q)φ(θ).
(5.14)
We obtain the following assertion.
Theorem 5.2. For any q ∈(0, 1) and RV X ≥0, the RV W R with CDF FW R and CF
ψ as in (5.13) and (5.14) is a proper RV with values in [0, ∞).
As an example, we next consider the case X taking integer values.
Example 5.1. We start with the simplest scenario: X = 1. For simplicity, suppose that
W0 = 0. In the model without resetting, the waiting time Wn = n →∞as n →∞. Cf.
(5.1). In the presence of resetting, we have a DTMC {W R
n} on the state space cZ+, with
transition probabilities P = (Pij): Pi,0 = q and Pi,i+1 = 1 −q for i ≥0. The stationarity
condition πP = π gives q P
i≥1
πi = π0 and (1 −q)πi = πi+1 for i ≥0. Thus, we obtain that
πi = q(1 −q)i, i ≥0, i.e., the RV W R is geometric. It is straightforward that the CDF FW R
and CF ψ satisfy (5.13) and (5.14).
Example 5.2.
Next, assume that X takes values k = 1, 2, . . ., with probabilities pk
and take again W0 = 0. Then the DTMC {W R
n} has transition probabilities Pi,0 = q for
i ≥0, and Pi,j = (1 −q)pj−i for i ≥0 and j ≥i + 1; the remaining entries equal to
zero. Let π = (π0, π1, . . . ) be the stationary distribution. Then we get q
∞
P
i=1
πi = π0, and
(1 −q)
iP
j=0
πjpi−j = πi for i ≥1. From this we obtain the equation for the characteristic
functions: ψ(θ) = 1 + (1 −q)ψ(θ)φ(θ), which results in (5.14).
6. The GI/GI/r queue with random resetting at arrival times
6.1. The Kiefer–Wolfowitz recursion for a GI/GI/r queue. In a standard GI/GI/r queue
with r > 1 servers and under the FCFS discipline, we operate with a collection of random
vectors {W n} where W n = (Wn1, . . . , Wnr) and 0 ≤Wn1 ≤. . . ≤Wnr. In other words, W n
takes values in the simplex S+
≤⊂Rr where S+
≤=
n
x = (x1, . . . , xr) : 0 ≤x1 ≤. . . ≤xr
o
.
Pictorially, W n represents the residual workload vector at the time of arrival of the nth job,
and its smallest entry, Wn1, gives the waiting time for the nth job.
The recursion that generates the sequence {W n} is due to Kiefer and Wolfowitz [15]:
W n+1 =
h
R
 W n + Vn e(1)
−Un 1
i
+
(6.1)
with the following ingredients on the RHS:

13
(i) Vn is the service time of the nth arrival, and Un is the time between the nth and
(n + 1)st arrival. It is assumed that the pairs (Un, Vn), n = 0, 1, . . ., form an IID
sequence. The joint CDF for (Un, Vn) is denoted by G:
G(u, v) = P(Un ≤u, Vn ≤v).
(6.2)
We will assume that CDF G is proper, i.e., RVs Un and Vn take ﬁnite values only.
(ii) e(1) = (1, 0, . . . , 0) ∈Zr
+ and 1 = (1, . . . , 1) ∈Zr
+ are r-dimensional 0, 1-vectors.
(iii) R
 W n + Vn e(1)
∈S+
≤is the result of the re-arrangement operation R applied to
the vector W n + Vn e(1) ∈Rr
+: the vector R
 W n + Vn e(1)
has the same collection
of entries as W n + Vn e(1) re-arranged in the non-decreasing order.
(iv)
h
R
 W n + Vn e(1)
−Uk 1
i+
∈S+
≤is the vector obtained when the negative entries
in R
 W n + Vn e(1)
−Uk 1 are replaced with zeros and non-negative entries are left
intact.
Equation (6.1) generates a DTMC {W n, n = 0, 1, . . .} on S+
≤. It can be re-written in
terms of the r-dimensional CDFs Fn(x) = P(W n ≤x), n ≥0, as follows:
Fn+1(x) =
Z
R2
Z
Sr 1

w ∈A(x, u, v)

dFn(w)dG(u, v),
x ∈Rr,
(6.3)
where the set A(x, u, v)) ⊂S+
≤is given by
A(x, u, v) =

w ∈S+
≤:
h
R(w + ve(1)) −u i
i+
≤x

(6.4)
and G(u, v) is given in (6.2). Here and below, the inequality between vectors means the
inequality between their respective entries.
As before, it is instructive to write equation (6.3) in an operator form
Fn+1 = KFn
where operator K acts on a CDF H by
(KH)(x) =
Z
R2
+
Z
S+
≤
1

w ∈A(x, u, v)

dH(w)dG(u, v),
x ∈Rr.
(6.5)
It is known that if the traﬃc intensity ρ :=
E[V ]
rE[U] < 1, then the stationary Kiefer–
Wolfowitz equation
W ≃
h
R
 W + V e(1)
−U i
i
+
or, equivalently, F = KF, i.e.,
F(x) =
Z
R2
Z
Sr 1(w ∈A(x, u, v))dF(w)dG(u, v)
(6.6)
has a unique solution giving a proper CDF F on Rr. In fact, for ρ < 1, the DTMC {W n} is
Harris ergodic. On the other hand, when ρ ≥1, there is no proper CDF F on Rr satisfying
equation (6.6), Cf. [2, Chapter XII.2].

14
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
6.2. The modiﬁed Kiefer–Wolfowitz recursion for a GI/GI/r queue with resetting. The model
with random resetting at arrival times again involves the parameter q ∈[0, 1). Set 0 =
(0, . . . , 0). Equations (6.1) and (6.3) are replaced with
W R
n+1 =



0 ,
with probability q,
h
R
 W R
n + Vn e(1)
−Un 1
i
+,
with probability 1 −q,
(6.7)
and
F R
n+1(x) = q1S+
≤(x) + (1 −q)
Z
R2
Z
S+
≤
1

w ∈A(x, u, v)

dF R
n(w)dG(u, v),
(6.8)
respectively, with F R
n(x) = P(W n ≤x).
As above, equation (6.7) determines a DTMC
{W R
n, n = 0, 1, . . .} on S+
≤. Again, the vector W R
n represents the residual workloads at the
servers at the nth arrival time.
Accordingly, the stationary equations for {W R
n} take the following equivalent forms:
(i)
W R ≃



0 ,
with probability q,
h
R
 W R + V e(1)
−U 1
i
+,
with probability 1 −q,
or
(ii)
F R(x) = q1S+
≤(x)
+(1 −q)
Z
R2
Z
S+
≤
1

w ∈A(x, u, v)

dFW Rn(w)dG(u, v),
or
(iii)
F R = q1S+
≤+ (1 −q)KF R
⇐⇒
 I −(1 −q)K

F = q1S+
≤
solved by
(iv)
F = q
 I −(1 −q)K
−11S+
≤
= q1S+
≤+ q(1 −q)K1S+
≤+ q(1 −q)2K21S+
≤+ . . . .
(6.9)
Proposition 6.1. For any q ∈(0, 1) and a sequence of
IID
RV pairs
{(Un, Vn)}, the
DTMC {W R
n} is Harris ergodic.
Proof. We will only give here a sketch of the (rather tedious) proof as it does not contain
serious novel elements. It is reduced to a repetition of arguments from [2, Chapters XII.1,
XII.2].
The crux of the matter is Theorem 1.2 on page 432 in [2, Chapter XII.1] and
Theorem 2.2 on page 345 in [2, Chapter XII.2] rewritten in a modiﬁed form for the DTMC
with resetting {W R
n}. In turn, the proof of the modiﬁed theorems is based on analogs of
Lemma 1.3 and Lemmas 2.3 and 2.4 in [2, Chapters XII.1, XII.2]. Such analogs connect the
DTMC {W R
n} with the majorizing Markov chain {f
W R
n} where arriving jobs are directed to
servers in the cyclic order with probability 1 −q and trigger resetting of the whole vector
of residual workloads to 0 with probability q. The analysis of the majorizing DTMC {f
W R
n}
is essentially reduced to the GI/GI/1 model with resetting which leads to the assertion of
Proposition 6.1.
□
The above construction then leads to the following result.

15
Theorem 6.1. For any q ∈(0, 1), the series in (6.9)(iv) determines a proper CDF satisfying
(6.9)(ii). Furthermore, equation (6.9)(ii) has a unique bounded solution, and this solution is
given by the series in (6.9)(iv).
Proof. As in the case of the model GI/GI/1 with resetting, the fact that the series in (6.9)(iv)
gives a solution to (6.9)(ii) follows from the construction with the help of Proposition 6.1.
Uniqueness is also established by the same argument as for the GI/GI/1 model.
□
7. The GI/GI/∞queue with random resetting at arrival times
7.1. The recursion for a GI/GI/∞queue. A standard GI/GI/∞can be described via a
sequence of random vectors W n = (Wn1, . . . , Wns), n = 0, 1, . . ., of a variable dimension
s = 0, 1, . . ., with entries Wn1 ≥. . . ≥Wns > 0 for s ≥1; for s = 0, one formally sets
W n = 0. Pictorially, W n represents the residual workload vector at the time of arrival of
the nth job, and its largest entry, Wn1, gives the time needed for clearing the system of jobs
entered before the nth arrival time. The equality W n = 0 means that the nth job ﬁnds an
empty queue at its arrival. Formally, W n takes values in the union O+(≥) := ∪
s≥0 Os
+(≥) of
simplexes Os
+(≥) =

x = (x1, . . . , xs) : x1 ≥. . . ≥xs > 0
	
of varying dimension s ≥1, and
a single-state set O0
+(≥) = {0} for s = 0.
The recursion for the residual workload vector {W n} in the GI/GI/∞model is given by
W n+1 = R

S
n
P(Vn, W n) −Un 1 s(n)+1

+
o 
,
n = 0, 1, . . . .
(7.1)
Here the RHS contains the following components:
(i) As in Section 6, Vn is the service time of the nth arrival, and Un is the time between
the nth and (n+1)st arrival. It is assumed that the pairs (Un, Vn), n = 0, 1, . . ., form
an IID sequence. The joint CDF for (Un, Vn) is again denoted by G and assumed to
be proper.
(ii) s(n) is the dimension of W n and the vector 1 s(n)+1 = (1, . . . , 1) ∈Zs(n)+1
+
has all
entries 1.
(iii) P(Vn, W n) ∈Rs(n)+1 is the result of concatenation of the value Vn and the vector
W n.
(iv)

P(Vn, W n)−Un 1 s(n)+1

+ is the vector obtained from P(Vn, W n)−Un 1 s(n)+1 when
negative entries are replaced with zeros and non-negative entries are left intact.
(v) S
n
P(Vn, W n) −Un 1 s(n)+1

+
o
is the result of shortening vector

P(Vn, W n) −
Un 1 s(n)+1

+ by removing the zero entries.
The dimension of S
n
P(Vn, W n) −
Un 1 s(n)+1

+
o
equals s(n + 1).

16
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
(iv) R

S
n
P(Vn, W n) −Un 1 s(n)+1

+
o 
is the result of the re-arrangement applied to
the vector S
n
P(Vn, W n) −Un 1 s(n)+1

+
o
. The vector R

S
n
P(Vn, W n) −Un 1 s(n)+1

+
o 
has the same collection of entries as S
n
P(Vn, W n) −Un 1 s(n)+1

+
o
re-arranged in
the non-increasing order.
Equation (7.1) generates a DTMC {W n, n = 0, 1, . . .} on O+(≥). The probability distri-
bution of W n on O+(≥) is described by a sequence F n = (Fn,0, Fn,1, Fn,2, . . .) of marginal
(non-normalized) CDFs Fn,k, where
Fn,0 = P
 W n = 0

, Fn,k(x(k)) = P
 s(n) = k, W n ≤x(k)
, x(k) ∈Rk, k ≥1.
(7.2)
For convenience, we will still refer to Fn,k as a CDF.
Equation (7.1) can be re-written in terms of the sequences F n, as follows:
Fn+1,0 = P
l
Z
R2
+
Z
Ol
+(≥)
1

w ∈A0,l(u, v)

dFn,l(w)dG(u, v),
Fn+1,k(x(k)) = P
l
Z
R2
+
Z
Ol
+(≥)
1

w ∈Ak,l(x(k), u, v)

dFn,l(w)dG(u, v),
x(k) ∈Rk, k ≥1.
(7.3)
Here, the sets A0,l(u, v), Ak,l(x(k), u, v)) ⊂Ol
+(≥) are given by
A0,l(u, v) =

w(l) ∈Ol
+(≥) : R

S
n
P(v, w(l)) −u 1 l+1

+
o 
= 0

,
Ak,l(x(k), u, v) =

w(l) ∈Ol
+(≥) : R

S
n
P(v, w(l)) −u 1 l+1

+
o 
∈Ok
+(≥),
R

S
n
P(v, w(l)) −u 1 l+1

+
o 
≤x(k)

,
k ≥1.
(7.4)
As before, it is instructive to write equation (7.3) in an operator form
Fn+1,k = P
l
Kk,lFn,l,
where operator Kk,l acts on a CDF Hl on Rl by
K0,lHl =
Z
R2
+
Z
Ol
+(≥)
1

w(l) ∈A0,l(u, v)

dH(w(l))dG(u, v),
(Kk,lHl)(x(k)) =
Z
R2
+
Z
Ol
+(≥)
1

w(l) ∈Ak,l(x(k), u, v)

dH(w(l))dG(u, v),
x(k) ∈Rk, k ≥1.
(7.5)
For future use, it is convenient to introduce the operator K = (Kk,l) with blocks Kk,l acting
on the CDF sequences H = (H0, H1, H2, . . .):
K H =

( K H)0 , ( K H)1 , ( K H)2 , . . .

where
( K H)k = P
l
Kk,lHl,
k = 0, 1, . . . .
(7.6)

17
7.2. The recursion for a GI/GI/∞queue with resetting. The recursion for a GI/GI/∞model
with resetting takes the form
W R
n+1 =



0,
with probability q,
R

S
n
P(Vn, W n) −Un 1 s(n)+1

+
o 
,
with probability 1 −q.
(7.7)
It generates a DTMC {W R
n, n = 0, 1, . . .} on O .
As above, we rewrite equation (7.7) in terms of the CDF F R
n = (F R
n,0, F R
n,1, . . .), where
F R
n,k = P

W R
n ∈Ok
+(≥), W n ≤x(k)
.
Then, we have
F R
n+1 = q 1 O +(≥) + (1 −q) K F R
n,
or, entry-wise,
F R
n+1,k(x(k)) = q1Ok
+(≥)(x(k)) + (1 −q) P
l
(Kk,lF R
n,l)(x(k)),
x(k) ∈Rk,
k = 0, 1, . . . .
(7.8)
The stationary version becomes
F R = q 1 O +(≥) + (1 −q) K F R ⇐⇒( I −(1 −q) K )F R = q 1 O +(≥)
or, entry-wise,
F R
n+1,k(x(k)) = q1Ok
+(≥)(x(k)) + (1 −q) P
l
(Kk,lF R
l )(x(k)),
k = 0, 1, . . . .
(7.9)
Equation (7.9) is solved by
F R = q

1 O +(≥) + (1 −q) K 1 O +(≥) + (1 −q)2 K 2 1 O +(≥) + . . .

(7.10)
Therefore we conclude the following result.
Theorem 7.1. The DTMC {W R
n} has a stationary probability distribution characterized
by equation (7.10).
8. Concluding Remarks
In this paper, we have considered the standard queueing models with random resettings.
Several extensions are possible future works. First, an immediate extension would be to
consider more general Markov chains with random resettings. It would be interesting to
identify conditions under which an explicit stationary distribution could be derived. Some
eﬀorts in this direction are made in our forthcoming paper [21]. Second, for non–Markovian
queues, it would be interesting to consider diﬀerent forms of resettings other than those
at arrival times. Third, diﬀusions have been established to approximate the performances
of queues in heavy traﬃc. It would be also worth considering such diﬀusion models with
random resetting, particularly, their ergodic properties and characterization of stationary
distributions.

18
GUODONG PANG, IZABELLA STUHL, AND YURI SUHOV
Acknowledgements.
G. Pang is partly supported by NSF grant DMS 2216765.
I.
Stuhl and Y. Suhov thank Math Dept, Penn State University for support. Y. Suhov thanks
DPMMS, University of Cambridge, and St John’s College, Cambridge, for support. Y. Suhov
thanks IHES, Bures-sur-Yvette, for hospitality during a visit in 2024.
References
[1] M. Abundo. The ﬁrst-passage area of Wiener process with stochastic resetting. Methodology and Com-
puting in Applied Probability, 25(4):92, 2023.
[2] S. Asmussen. Applied Probability and Queues. Springer-Verlag, 2003.
[3] W.
B¨ohm.
A
note
on
queueing
systems
exposed
to
disasters.
Working
paper,
2008.
https://research.wu.ac.at/en/publications/a-note-on-queueing-systems-exposed-to-disasters-3.
[4] O. L. Bonomo, A. Pal and S. Reuveni. Mitigating long queues and waiting times with service resetting.
PNAS nexus, 1(3), p.pgac070, 2022.
[5] P.C. Bressloﬀ. Queueing theory of search processes with stochastic resetting. Physical Review E., 102(3),
p.032109, 2020.
[6] O. J. Boxma, D. Perry, and W. Stadje. Clearing models for M/G/1 queues. Queueing Syst., 38:287–306,
2001.
[7] G. S. Bura. Transient solution of an M/M/∞queue with catastrophes. Communications in Statistics-
Theory and Methods, 48(14):3439–3450, 2019.
[8] S. Dharmaraja, A. Di Crescenzo, V. Giorno, and A. G. Nobile. A continuous-time Ehrenfest model with
catastrophes and its jump-diﬀusion approximation. Journal of Statistical Physics, 161:326–345, 2015.
[9] A. Di Crescenzo, V. Giorno, B. Krishna Kumar, and A. G. Nobile. A double-ended queue with catas-
trophes and repairs, and a jump-diﬀusion approximation. Methodology and Computing in Applied Prob-
ability, 14:937–954, 2012.
[10] A. Di Crescenzo, V. Giorno, A. G. Nobile, and L. M. Ricciardi. On the M/M/1 queue with catastrophes
and its continuous approximation. Queueing Systems, 43:329–347, 2003.
[11] S. Dimou and A. Economou. The single server queue with catastrophes and geometric reneging. Method-
ology and Computing in Applied Probability, 15:595–621, 2013.
[12] V. Giorno, A. G. Nobile, and R. di Cesare. On the reﬂected Ornstein–Uhlenbeck process with catastro-
phes. Applied Mathematics and Computation, 218(23):11570–11582, 2012.
[13] V. Giorno, A. G. Nobile, and S. Spina. On some time non-homogeneous queueing systems with catas-
trophes. Applied Mathematics and Computation, 245:220–234, 2014.
[14] J.F.C. Kingman. On the algebra of queues. J. Appl. Probability, 3:285–326, 1966.
[15] J. Kiefer and J. Wolfowitz. On the theory of queues with many servers. Transactions of the American
Mathematical Society, 78(1):1–18, 1955.
[16] B. Krishna Kumar, A. Krishnamoorthy, S. Pavai Madheswari, and S. Sadiq Basha. Transient analysis
of a single server queue with catastrophes, failures and repairs. Queueing Systems, 56:133–141, 2007.
[17] Lindley, D. V. The theory of queues with a single server. Math. Proc. Cambridge Phil. Soc. 48(2):
277–289, 1952.
[18] T. M. Michelitsch, G. d’Onofrio, F. Polito, and A. P. Riascos. Random walks with stochastic resetting
in complex networks: a discrete time approach. arXiv:2409.08394, 2024.
[19] G. C. Mytalas and M. A. Zazanis. An M X/G/1 queueing system with disasters and repairs under a
multiple adapted vacation policy. Naval Research Logistics, 62(3):171–189, 2015.
[20] A. Pal, V. Stojkoski and T. Sandev. Random resetting in search problems. In Target Search Problems,
pp. 323-355, 2024. Cham: Springer Nature Switzerland.
[21] G. Pang, I. Stuhl and Y. Suhov. Markov chains with random resetting. In preparation, 2025.

19
[22] R. Roy, A. Biswas and A. Pal. Queues with resetting: a perspective. Journal of Physics: Complexity,
5(2), p.021001, 2024.
[23] R. Serfozo and S. Stidham. Semi-stationary clearing processes. Stoch. Proc. Appl., 6(2):165–178, 1978.
[24] S. Stidham Jr. Stochastic clearing systems. Stoch. Proc. Appl., 2(1):85–113, 1974.
[25] V. Stojkoski, T. Sandev, L. Kocarev, and A. Pal. Geometric Brownian motion under stochastic resetting:
A stationary yet nonergodic process. Physical Review E, 104(1):014121, 2021.
[26] R. Sudhesh and S. Dharmaraja. Analysis of a multiple dual-stage vacation queueing system with disaster
and repairable server. Methodology and Computing in Applied Probability, 24(4):2485–2508, 2022.
[27] D. Vinod, A. G. Cherstvy, R. Metzler, and I. M. Sokolov. Time-averaging and nonergodicity of reset
geometric Brownian motion with drift. Phys. Rev. E, 106(3):034137, 2022.
[28] W. Whitt. The stationary distribution of a stochastic clearing process. Operations Research, 29(2):294–
308, 1981.
[29] M. Yajima and T. Phung-Duc. A central limit theorem for a Markov-modulated inﬁnite-server queue
with batch Poisson arrivals and binomial catastrophes. ACM SIGMETRICS Performance Evaluation
Review, 46(3):33–34, 2019.
∗Department of Computational Applied Mathematics and Operations Research, George
R. Brown School of Engineering and Computing, Rice University, Houston, TX 77005
Email address: gdpang@rice.edu
†Department of Mathematics, Penn State University, University Park, PA 16802
Email address: ius68@psu.edu
$Department of Mathematics, Penn State University, University Park, PA 16802; DP-
MMS, University of Cambridge, Cambridge CB3 0WB, UK; St John’s College, Cambridge,
Cambridge CB2 1TP, UK
Email address: ims14@psu.edu, yms@statslab.cam.ac.uk
