Mathematical Problems in Engineering
two-hop architecture, radio signals do not need to penetrate
into the carriages, and thus the radio signal penetration loss
problem is resolved.
Distributed content servers are deployed in order to
ooad data trac from the backbone network []. When a
service is requested from the passengers, the data packets can
be fetched from the corresponding content server (CS). To
make the analysis of such network tractable, we assume each
CS can provide one type of service. Notice that this can be
easily extended to the situation where each CS can provide
multiple types of services while one buer is allocated to
each type of service in the CS, which will be clear later on in
Section .. In order to simplify the protocol design for HSR
applications, erasure coding based service delivery is consid-
ered [] and the advantage is that no retransmission scheme
is required for the transmission error due to a highly dynamic
wireless channel condition.
Compared to the traditional cellular networks, the deter-
ministic train trajectory in HSR networks is a unique feature
[, ]. e train trajectory represents the location of a train at
a specic time, and BS provides the service delivery if the
train is under its coverage. Since the train moves on a pre-
determined rail line and the velocity is relatively steady, the
information of train trajectory can be obtained in advance
with high accuracy so that the service packets can be
delivered by the specic BS at a certain time. erefore, this
paper focuses on the delay-aware service scheduling problem
regardless of which BS is used for service delivery.
2.1. Physical-Layer Model. We consider a time-slotted system
for downlink service transmission with slot period
𝐵
.When
the train moves along the railway, BS and AP can transmit
simultaneously without interference by operating on dierent
frequency bands. For the communication in the BS-RS link,
we consider the MAC frame structure proposed in []which
is specically designed for high-speed trains with a speed of
up to km/h. For the communication in the AP-users link,
traditional WLAN standards, for example, IEEE .a/b/g,
are employed since passengers within the train are relatively
stationary with respect to the AP.
Let H()=H
1
()
H
2
()be the joint CSI, where H
1
()=
{
BS,𝑅
(),∀}with the channel gain
BS,𝑅
in the BS-RS link
and H
2
()={
𝑅,𝑘
(),∀}with the channel gain
𝑅,𝑘
between
the AP and the user requesting service .echannelfrom
BS to the RS is assumed to follow the Rician distribution [],
while the channel between AP and the users can be treated as
Rayleigh fading channel []. We assume that the channel
gains remain constant during a slot duration and change
acrossslotsinani.i.d.manner.Inaddition,toenhancetrans-
mission performance, the strong channel coding is used, and
hence, the maximum achievable data rate is achieved by the
instantaneous mutual information. For the BS-RS link with
the bandwidth
1
, the maximum achievable data rate in bit-
per-second is given by
BS
=
1
log
2
1+
BS
BS,𝑅
2
0
,
()
where
BS
is the transmit power of the BS and
0
is the noise
power. Since all the services share the common channel in the
BS-RS link, they have the same data rate
BS
.Likewise,forthe
AP-users link with the bandwidth
2
,wehavethemaximum
achievable data rate for service
𝑅,𝑘
=
2
log
2
1+
𝑅
𝑅,𝑘
2
0
, =1,2,...,,
()
where
𝑅
is the transmit power of the AP. In addition, we
assume the packets have equal size of bits, the capacity
BS
can be expressed by the maximum number of packets by con-
verting
BS
through a proportionality constant; that is,
BS
=
BS
𝐵
/.Similarly,
𝑅,𝑘
=
𝑅,𝑘
𝐵
/,∀.
2.2. Packet-Level Model. Assume that there are types of
services requested by the passengers. Each type of service is
allocated with two buers, one in the th CS denoted by Q
CS,𝑘
and the other one in the RS denoted by Q
𝑅,𝑘
,for=1,...,.
e two buers for each service in the CSs and RS are in
tandem. Let
𝑄
be the maximum number of packets for the
buers in the RS and all CSs. Let Q() = Q
1
()
Q
2
()be
the joint QSI, where Q
1
() = {
CS,𝑘
(),∀} and Q
2
() =
{
𝑅,𝑘
(),∀}. Specically,
CS,𝑘
() and
𝑅,𝑘
() denote the
number of packets at the beginning of slot in the buer of
th CS and the th buer in RS, respectively.
ese heterogeneous services have dierent packet
arrival rates and QoS requirements. Data packets from the
higher layer application arrive into the buers and are queued
until they are transmitted. e packet arrival process for each
typeofserviceisassumedtobei.i.d.acrossslots.Let
𝑘
()
denote the number of packets arriving into the buer in th
CS at slot . Suppose in general,
𝑘
()follows the truncated
Poisson distribution
𝑘
() with average packet arrival rate
𝑘
= E[
𝑘
()]for service .edistribution
𝑘
()can be
given as
𝑘
(
)
=exp −
𝑘
𝑏
𝑘
!
, =0,...,
𝑘
,
()
where
𝑘
is found assuming
𝑘
(
𝑘
)→0.
e average number of arriving packets at the th CS is
given by
𝑘
(1−
𝑑
𝑘
),where
𝑑
𝑘
is the dropping probability. is
is the same as the average number of packets received by
the corresponding buer in RS since the two buers are in
tandem. e suciently large buer and negligible dropping
probability (
𝑑
𝑘
≈0) assumptions are considered in this paper.
2.3. MDP Model. In this paper, the service scheduling prob-
lem for the two-hop link in HSR networks is formulated as an
innite-horizon average cost CMDP. To make the analysis of
the CMDP problem tractable in the sequel, it is necessary to
identify the elements of MDP model in our scheduling prob-
lem. In general, an MDP model consists of ve elements: deci-
sion epochs, states, actions, cost function, and state transition
probability function. We describe these elements as follows.
e scheduling decisions for the data packet delivery in
the two-hop link have to be made slot by slot and the