#### On a tandem queue with batch service and its applications in wireless sensor networks

On a tandem queue with batch service and its applications in wireless sensor networks
Mihaela Mitici 0 1 2
Jasper Goseling 0 1 2
Jan-Kees van Ommeren 0 1 2
Maurits de Graaf 0 1 2
Richard J. Boucherie 0 1 2
0 Stochastic Operations Research, Department of Applied Mathematics, University of Twente , P.O. Box 217, 7500 AE Enschede , The Netherlands
1 Thales Nederland B.V. , Bestevaer 46, 1271 ZA Huizen , The Netherlands
2 Air Transport and Operations, Delft University of Technology , P.O. Box 5058, 2600GB Delft , The Netherlands
We present a tandem network of queues 0, . . . , s − 1. Customers arrive at queue 0 according to a Poisson process with rate λ. There are s independent batch service processes at exponential rates μ0, . . . , μs−1. Service process i , i = 0, . . . , s−1, at rate μi is such that all customers of all queues 0, . . . , i simultaneously receive service and move to the next queue. We show that this system has a geometric product-form steady-state distribution. Moreover, we determine the service allocation that minimizes the waiting time in the system and state conditions to approximate such optimal allocations. Our model is motivated by applications in wireless sensor networks, where s observations from different sensors are collected for data fusion. We demonstrate that both optimal centralized and decentralized sensor scheduling can be modeled by our queueing model by choosing the values of μi appropriately. We quantify the performance gap between the centralized and decentralized schedules for arbitrarily large sensor networks.
Tandem network of queues with Batch Service; Wireless Sensor Networks; Broadcasting; Scheduling
1 Introduction
We consider a tandem network of queues 0, . . . , s − 1. Customers arrive
one-byone at queue 0 according to a Poisson process with rate λ. There are s independent
batch service processes at exponential rates μ0, . . . , μs−1. Service process i , i =
0, . . . , s − 1, at rate μi is such that all customers of all queues 0, . . . , i simultaneously
receive service and move to the next queue.
This system does not satisfy partial balance in classical form at queue 0, since
customers arrive one-by-one and are served in batches. Therefore, our system cannot
fall in the class of Kelly–Whittle networks [12] and it does not fall in the class of batch
routing queueing networks with product form [6,14].
In isolation, queue 0 can be modeled as a queue with disasters [9]; customers arrive
one-by-one and the entire queue is emptied according to a Poisson process of triggers
with rate μ0 + · · · + μs−1. However, our tandem network of queues cannot be modeled
as a network with triggers or negative customers (see, for example, [5] for a description
of these networks). The reason is that in a tandem network with triggers or negative
customers that empty the entire queue, a trigger or a negative customer that finds a
queue empty is lost. In our system the contents of queues 0, . . . , i are shifted to queues
1, . . . , i + 1 also when some of the queues 0, . . . , i are empty.
Our model is motivated by applications in data collection from a wireless sensor
network (WSN). In particular, we investigate the case where clients arriving at the
network are interested in collecting data from different sensors for data fusion. These
sensors broadcast their data, i.e., all clients receive the data of a sensor when it is
transmitted. A client needs to obtain data from an arbitrary set of s ∈ N different sensors
in order to be able to apply a fusion algorithm. Examples of such applications are i)
localization for client positioning [3]; ii) the retrieval of various noisy measurements
of the same attribute for data fusion [16].
Data fusion in WSNs has been studied extensively in, for instance, [2,17,19–21].
Scheduling for WSNs has been extensively studied in, for example, [1,11]. However,
little work has been done on sensor transmission schedules that support data fusion
[16,18], as aimed at here. Sensor transmission schedules affect the time for a client to
receive sufficient data from distinct sensors in order to be able to apply a data fusion
algorithm. In this work, we consider a tandem queueing model and we show it can
be used to analyze sensor transmission schedules under which clients collect a fixed
number of sensor observations to be able to apply data fusion algorithms.
The problem of providing a fixed number of units of service (in our case, a set
of s observations) to all clients has been studied in, for instance, [8,13,15]. In [8] a
discrete-time multi-server queueing model with a general arrival process is considered,
where each client is interested in receiving s units of service. Contrary to our model, in
each time slot each client is guaranteed to receive a unit of service. In [15] a
discretetime queue is considered with geometric arrivals and arbitrarily distributed service
requirements (number of units of service required). Contrary to our model, a single
class of customers and a single server is considered. Our model has similarities to
gated service polling models [4]. In particular, we could place customers of each class
at different queues and provide gated batch service [7] at one of these queues.
The contributions of this paper are as follows. We model our system as a multi-class
tandem network of queues with batch service and demonstrate that our system has a
geometric product-form steady-state distribution. Moreover, we determine the service
allocation strategies that minimize the expected waiting time in the system and state
conditions to approximate such optimal allocations. We also provide a closed form
expression for the Hellinger distance to an optimal allocation. We show that this
queueing model has applications in data collection from a WSN where the class of a
client in the queueing model corresponds to the number of observations that a client
has already collected from a WSN. Analyzing a scheduling mechanism for the WSN
data collection application now reduces to analyzing this multi-class queue under a
specific assignment of service rates for these classes. As special cases, we consider
a decentralized and an optimal centralized broadcasting schedule and determine the
performance gap between the two with respect to waiting time, for arbitrarily large
WSNs. As such, this paper introduces a novel product-form queueing model that is of
theoretical interest in itself and has interesting practical applications to WSNs.
The remainder of this paper is organized as follows. In Sect. 2 we formulate the
model and introduce some notation. In Sect. 3 we analyze our model and obtain the
steady-state distribution. In Sect. 4 we determine optimal service allocations with
respect to waiting time and state conditions to approximate such allocations.
Applications to sensor networks are provided in Sect. 5. In Sect. 6 we discuss the results and
provide conclusions.
2 Model and notation
We consider a tandem network of queues 0, . . . , s − 1 in which there are s customer
classes, labeled 0, . . . , s −1. Customers arrive according to a Poisson process with rate
λ and have class 0. There are s independent batch service processes at exponential
rates μ0, . . . , μs−1. Service process i , i = 0, . . . , s − 1, at rate μi is such that all
customers of all classes 0, . . . , i simultaneously receive service (but not those of class
i + 1, . . . , s − 1). Customers of classes 0, . . . , s − 2 that receive service increase their
class by 1 and remain in the system. Customers of class s − 1 that receive service leave
the system.
We will describe the continuous-time Markov chain, X = (X (t ), t ≥ 0), that
captures the dynamics of this queueing model, but before doing so we introduce the
following notation. Let N0 = {0, 1, . . . }. For n ∈ Ns0 we denote its elements through
subscripts, i.e., n = (n0, . . . , ns−1). Next, consider s functions Ui : Ns0 → Ns0,
0 ≤ i ≤ s − 1, where Ui (n) = (0, n0, . . . , ni−1, ni + ni+1, ni+2, . . . , ns−1) for
i = 0, . . . , s − 2, and Us−1(n) = (0, n0, . . . , ns−2).
We have a continuous-time Markov chain X on the state space Ns0 in which n ∈ Ns
0
represents the state in which there are ni customers of class i , i = 0, . . . , s − 1, in the
system. The outgoing transitions from state n are as follows. For each 0 ≤ i ≤ s − 1 a
transition occurs from n to Ui (n) at a rate μi . Also, there is a transition at rate λ from
n to n + e0, where e0 = (1, 0, . . . , 0) is of length s.
If n contains zeros, some care is required with the above definition of the transition
rates. We illustrate this by means of two examples. Firstly, let 0 < i ≤ s − 1 and
ni−1
ni−2
i − 1
ni−1
ni+1 − x
Fig. 1 States of the general continuous-time Markov chain
consider n j = 0 iff j = i . In this case Ui (n) = Ui−1(n). Therefore, the transition rate
from n to Ui−1(n) is μi−1 + μi . Secondly, let 0 < i ≤ s − 1 and consider n j = 0 iff
j ≤ i . In this case U0(n) = U1(n) = · · · = Ui (n) = n. Note that we do not have a
transition from n to U j (n) for j ≤ i in this case.
For notational convenience in the next section, we introduce the functions Si :
Z × Zs → Zs0, for i = 0, . . . , s − 1. These functions are defined as Si (x , n) =
(n1, . . . , ni , x , ni+1 − x , ni+2, . . . , ns−1) for i = 0, . . . , s − 2 and Ss−1(x , n) =
(n1, . . . , ns−2, x ). The functions Ui and Si are illustrated in Fig. 1. Finally, let μ¯ i =
sk−=1i μk . This is the total rate at which customers of class i receive service and change
their class to i + 1. Note that μ¯ 0 = sk−=10 μk and μ¯ s−1 = μs−1.
3 Analysis
In this section, we discuss the balance equations of the Markov chain X and derive
the steady-state distribution π(n) (n ∈ Ns0) of the system in Sect. 2.
The reason for introducing Si in the previous section is that Si (x , n) provides a
convenient means to describe the states with a transitions into n. Indeed, ignoring
boundary conditions for the moment and considering n ∈ Zs with n0 = 0, we have
Ui (Si (x , n)) = n for all 0 ≤ i ≤ s − 1 and all x ∈ Z. In other words, for all
0 ≤ i ≤ s − 1 and x ∈ Z, Si (x , n) has a transition to n with rate μi . The one thing we
need to deal with is that Si (x , n) itself must be an element of the state space, i.e., we
need Si (x , n) ∈ Ns0. It is readily verified that for 0 ≤ i ≤ s − 2 this is true iff n ∈ Ns
0
and 0 ≤ x ≤ ni+1. Also, Ss−1(x , n) ∈ Ns0 iff n ∈ Ns0 and x ≥ 0.
Before we determine the steady-state distribution of this system, we first discuss
the balance equations. Let k = min{i | ni > 0, 0 ≤ i ≤ s − 1}. Note that, as discussed
in Sect. 2, Ui (n) = n for 0 ≤ i ≤ k − 1. Note, in addition, that Si (x , n) = n for
0 ≤ i ≤ k − 2 and 0 ≤ x ≤ ni+1. Also, Sk−1(0, n) = n. Now, for n0 = 0, the
resulting balance equation is
s−1
i=k
i=k x=0
Adding π(n)μi = nxi=+01 π(Si (x , n))μi = π(Si (0, n))μi for 0 ≤ i ≤ k − 2, as well
as π(n)μk−1 = π(Sk−1(0, n))μk−1 to the left-hand side as well as the right-hand side
of (3.1), (3.1), i.e., the balance equation for n0 = 0, becomes
s−1
i=0
s−2 ni+1
i=0 x=0
x=0
where we note that the exponent of the first i terms is n j+1.
We now consider the general case s > 1. First, we consider (3.2) for the (s −
1)dimensional system with service rates μ0, μ1, . . . , μs−3, μs−2 + μs−1. Note that
x=0
ni+1−x
We next determine the steady-state distribution of the Markov chain X .
Theorem 3.1 The steady-state probability distribution of the continuous-time Markov
chain X defined above is
For the case n0 > 0, it is readily verified that the balance equation is as follows:
s−1
i=0
x=0
s−1
i=0
ni s−1
where ni ≥ 0 for 0 ≤ i ≤ s − 1.
Proof First, it is readily verified that (3.4) satisfies (3.3) for n0 > 0.
For n0 = 0, we use induction on s to prove that (3.4) satisfies (3.2). As a base case,
we consider s = 1. For s = 1, by observing that in this case μ¯ 0 = μ0, (3.2) reduces
to
which is clearly satisfied by the geometric distribution (3.4).
Before considering the general case s > 1, for clarity, observe that, using (3.4), for
0 ≤ i ≤ s − 1,
j=i+2
n j s−1
μ¯ s−2 = μs−2 + μ¯ s−1. We assume that (3.2) for this (s − 1)-dimensional system has
the solution provided in (3.4). Based on this induction hypothesis, we will show that
(3.2) for an s-dimensional system has the solution provided in (3.4).
Now, writing (3.2) for this (s − 1)-dimensional system in detail with the
productform steady-state distribution according to (3.4) gives
s−2
i=0
s−3 ni+1
i=0 x=0
i−1
x=0 i=0
ni+1−x s−2
j=i+2
Note that, in (3.8), n is of length s. It remains to show that the second term of the
right-hand side of (3.8) equals
s−3 ni+1
i=0 x=0
s−3
i=0
ns−1
s−3
i=0
ns−1
The first term on the right-hand side of (3.9) is
ns−1
x=0
ns−1
x=0
s−3
i=0
ns−1
where in the last equation we used that μ¯ s−1 = μs−1.
It is easy to verify that (3.9) follows from (3.10) and (3.11). The proof that (3.4)
satisfies (3.2) now follows from (3.8) and (3.9).
From Little’s law we readily obtain that the expected waiting time for a customer
in a system with s customer classes is
From Theorem 3.1, we also readily obtain that the expected length of the busy
period of the system is
s−1
i=0
s−1
−1
where we used that μs−2 = μ¯ s−2 − μ¯ s−1 and we have evaluated the geometric sum
ns−1 x
λ + μ¯ s−2
x=0Similarly, the second term on the right-hand side of (3.9) is
x=0
s−2
i=0
s−2
i=0
ni+1 ∞
x=0
4 Optimal assignment of service rates
Consider the system introduced in Sect. 2. We further assume that the total service rate
μ is distributed over μi , i = 0, . . . , s − 1, i.e., μ = μ1 + . . . + μs−1. In this section,
we determine the service allocation that minimizes the expected waiting time, and the
conditions to approximate such optimal allocations. The following lemma formalizes
the intuitive fact that it is best to choose μs−1 as large as possible.
Lemma 4.1 The system with μi = 0 for 0 ≤ i < s − 1 and μs−1 = μ minimizes the
expected waiting time among all systems with the property that μ0 + . . . + μs−1 = μ.
Proof From (3.12), it immediately follows that the expected waiting time is minimized
when all μ¯ i take their maximal value. As μ¯ 0 = μ0 + . . . + μs−1 = μ, and μ¯ 0 ≥ μ¯ 1 ≥
. . . ≥ μ¯ s−1 = μs−1, this maximum is attained when μ¯ 0 = μ¯ 1 = . . . = μ¯ s−1 = μ,
which corresponds to the service rate assignment stated in the lemma.
Consider the optimal service rate assignment
From Lemma 4.1 it follows that the steady-state distribution π C of this system is as
follows. Here, we used the superscript C to indicate that to achieve such an optimal
rate assignment we would require central coordination.
Corollary 4.2 For ni ≥ 0, 0 ≤ i ≤ s − 1, the steady-state distribution of the system
under the optimal service rate assignment is
, n = 0,
with expected waiting time, denoted by WsC ,
We next determine the Hellinger distance [10] between a distribution π ,
corresponding to a general system with μi as defined in Sect. 2, and the optimal system
π C , as defined in Corollary 4.2. We denote the Hellinger distance between π1 and π2
by H (π1, π2), where
The Hellinger distance H (π1, π2) and the total variation distance, denoted by
δ(π1, π2), satisfy
The maximum distance 1 between the two distributions is achieved when π1 assigns
probability 0 to every set to which π2 assigns a positive probability, and vice versa.
From Theorem 3.1 and Corollary 4.2, it is readily verified that H (π, π C ) is as
follows:
1 −
s−1
i=0
We next investigate under which conditions an arbitrary system can have a Hellinger
distance approaching zero to the optimal system. To this end, consider the system
introduced in Sect. 2, with s independent batch service processes at exponential rates
μ0, . . . , μs−1. Assume μi depends on s and N , so μi = νi (s, N ), i = 0, . . . , s − 1,
where N is, for the moment, an arbitrary system parameter denoting the size of the
WSN.
Using the fact that μ¯ i = sk−=1i μi and Lemma 4.3, the next result follows.
Lemma 4.4 For any λ > 0, limN →∞ H (π, π C ) = 0 iff limN →∞ νi (s, N ) = 0 for
0 ≤ i < s − 1 and limN →∞ νs−1(s, N ) = μ.
Proof Note that H (π, π C ) is 0 only if the product in Lemma 4.3 tends to 1 in the
limit. As for each i = 0, . . . , s − 1 each term
the product tends to 1 only if each term individually tends to 1. By straightforward
calculations it follows that √μμ¯ i /√(λ + μ)(λ + μ¯ i ) − λ = 1 only if λ = 0 or
μ¯ i = μ. Thus, H (π, π C ) → 0 iff μ¯ i → μ for any i, 0 ≤ i ≤ s − 1. We next evaluate
under which conditions μ¯ i → μ, ∀i . For i = s − 1, it follows that μs−1 → μ. For
0 ≤ i < s − 1 this follows from the observation that μ = μ0 + . . . + μs−1 = μ¯ 0 ≥
μ¯ 1 ≥ . . . ≥ μ¯ s−1 = μ.
5 Applications in wireless sensor networks
In this section, we show that the problem of collecting a fixed number of sensor
observations from a WSN to be able, for instance, to apply data fusion algorithms, can
be modeled using the queueing system introduced in Sect. 2.
Consider a WSN consisting of N sensors. Clients arrive at the network according
to a Poisson process at rate λ and need to obtain s observations from distinct sensors
in order to apply a fusion algorithm. We assume that any set of s observations from
distinct sensors suffices and that there is a one-to-one correspondence between sensors
and observations, i.e, different sensors transmit different observations. The sensors
broadcast their data, i.e., all clients receive the data of a sensor when it is transmitted.
A transmission schedule determines which sensor transmits at which time. In the
remainder of this section, we analyze two different scheduling strategies and their
impact on the system’s performance. We also quantify the performance gap between
the two schedules with respect to waiting time, for arbitrarily large WSNs.
First we show that the WSN described above can be modeled using the queueing
system in Sect. 2.
Lemma 5.1 All clients that have obtained i different observations, 0 ≤ i ≤ s − 1,
have the same set of observations. Moreover, the observations of the clients that have
obtained i observations are a subset of the observations obtained by clients that have
j > i observations.
Proof Initially there are no clients in the network and the conditions are satisfied. The
proof directly follows from an induction over the number of events by considering
two possible events: i) arrival of a client and ii) transmission of an observation (which
is useful to all or part of the clients in the system).
It follows from Lemma 5.1 that we can identify customers of class 0 ≤ i ≤ s − 1 in
the queueing system with those clients that have obtained i different observations and
that are waiting to collect s − i additional observations. Indeed, if customers of class i
receive service (obtain a new observation), then also clients of classes 0, 1, . . . , i − 1
receive service, since their observations form a subset of those of class i customers.
It remains to quantify the values of the service rates μ0, . . . , μs−1. In the next two
subsections, we will do this for two specific broadcasting schedules.
5.1 Decentralized broadcasting and optimal broadcasting
First, we consider a decentralized broadcasting schedule, D, where each sensor
transmits independently of the other sensors at an exponential rate μD/N . Note that the
overall transmission rate of observations is μD.
Lemma 5.2 Under a decentralized broadcasting schedule
if 0 ≤ i < s − 1,
Proof For 0 ≤ i < s − 1, μi is the rate at which all customers of classes 0, . . . , i , but
no customers of classes i +1, . . . , s −1, receive a new observation. From Lemma 5.1 it
follows that this new observation must be exactly the one observation that has already
been received by customers of class i + 1, but not by customers of class i . Thus,
this rate corresponds to the rate at which one specific sensor is transmitting, which is
μD/N . If i = s − 1, all N − (s − 1) observations that have not yet been received by
customers of class s − 1 will cause them to increase their class. These observations
are transmitted by N − (s − 1) sensors that transmit independently at rate μD/N .
From Theorem 3.1 and Lemma 5.2 it follows that the steady-state distribution of
the system under the D schedule is as follows:
Corollary 5.3 For ni ≥ 0, 0 ≤ i ≤ s − 1, the steady-state distribution of the system
under the D schedule is
s−1
i=0
ni s−1
i=0
with expected waiting time, denoted by E[WsD],
s−1
i=0
Next we consider the following centralized broadcasting schedule C : at an
exponential rate μC an observation is broadcast from a sensor whose observation causes
all clients in the system to increase their class, i.e, the observation is broadcast by
a sensor that has not transmitted its observation to any of the clients present in the
network. One way to achieve an optimal schedule is to follow a round-robin schedule,
in which the N sensors are scheduled sequentially in a cyclic way. Another way is to
keep track of the sensors that have broadcast observations to the customers present in
the system and not schedule these sensors for transmission. As discussed in Sect. 4,
we have
From Lemma 4.3, Lemma 5.2 and (5.2) the next corollary follows.
Corollary 5.4 If μD = μC , then
1 −
s−1
i=0
μ√1 − i /N
√(λ + μ)(λ + μ(1 − i /N )) − λ
From Corollary 5.4, we readily have that for N → ∞, H (π D, π C ) → 0.
Figure 2 shows H (π D, π C ) for finite N . Under the parameters considered, the
distance H (π D, π C ) rapidly decreases as a function of the network size N .
Theorem 5.5 If μD = γ (s, N )μC , where
then E[WsD ] = E[WsC ].
If we jointly let N → ∞ and s →
s−1
i=0
i
1 − N
−1
where the result follows from evaluating lims→∞ s−1 0s−1(1 − x /(δs))−1d x .
Lastly, if s is kept constant while N → ∞, we have lim N →∞ γ (s, N ) = 1.
6 Conclusions
We have introduced a new type of multi-class tandem network of queues with batch
service and demonstrated that this system has a geometric product-form steady-state
distribution. Moreover, we have shown that in order to have an optimal service
allocation with respect to waiting time it is required to maximize the service rate of the last
queue in the tandem network of queues considered. We have shown that our queueing
model has applications in data collection in WSNs. As specific WSN applications, we
considered optimal centralized and decentralized sensor broadcasting schedules. We
have shown for which choices of service rates our queueing model is appropriate for
the two types of schedules. We have also determined the steady-state distribution of
the system under these two schedules. Lastly, we have characterized the performance
gap between the two schedules with respect to waiting time, for arbitrarily large sensor
networks.
Acknowledgements This work was performed within the project RRR (Realisation of Reliable and Secure
Residential Sensor Platforms) of the Dutch program IOP Generieke Communicatie, IGC1020, supported
by the Subsidieregeling Sterktes in Innovatie.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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