We consider a scheduling problem for a two-hop queueing network where the queues have randomly varying connectivity. Customers arrive at the source queue and are later routed to multiple relay queues. A relay queue can be served only if it is in connected state, and the state changes randomly over time. The source queue and relay queues are served in a time-sharing manner; that is, only one customer can be served at any instant. We propose Join the Shortest Queue-Longest Connected Queue (JSQ-LCQ) policy as follows: (1) if there exist nonempty relay queues in connected state, serve the longest queue among them; (2) if there are no relay queues to serve, route a customer from the source queue to the shortest relay queue. For symmetric systems in which the connectivity has symmetric statistics across the relay queues, we show that JSQ-LCQ is strongly optimal, that is, minimizes the delay in the stochastic ordering sense. We use stochastic coupling and show that the systems under coupling exist in two distinct phases, due to dynamic interactions among source and relay queues. By careful construction of coupling in both phases, we establish the stochastic dominance in delay between JSQ-LCQ and any arbitrary policy.
We consider a scheduling problem in queueing systems with random connectivity of servers. For example, in wireless communication systems, the communication channel may randomly become unavailable for data transmissions due to fluctuation of channel quality over time. To cover areas with poor channel quality, relay networks have been widely adopted [
We consider a queueing model depicted in Figure
Two-hop queueing network model. The system consists of one source queue (SQ) and
Potential engineering applications of our queueing model include wireless relay networks. In 5G communication systems, wireless relays are expected to be widely used to enhance capacity and coverage of the network [
The time-sharing service of customers between SQ and RQs is analogous to half-duplex transmissions of packets in wireless relay networks, that is, only either BS or RN can transmit at a time slot. While full-duplex relays are recently under investigation [
We introduce a policy called Join the Shortest Queue-Longest Connected Queue (JSQ-LCQ).
The JSQ-LCQ policy is defined as follows: If there exist RQs which are nonempty and connected, serve a customer from the If there is no RQ to serve, route a customer from the SQ to the
It is observed that the JSQ-LCQ is a simple and greedy policy with tehe following properties: JSQ-LCQ prioritizes serving the RQs over serving the SQ. If there is any chance to serve a RQ, it will do so. Among the connected RQs, it will serve the longest one. This is an attempt to make the RQs as “balanced” as possible. When JSQ-LCQ routes a customer to RQ, it chooses the shortest RQ. Again the policy attempts to balance the RQs.
JSQ-LCQ focuses on balancing queues during the service and routing so as to maximize the “opportunism” of time-varying connectivity, that is, so that as many nonempty queues as possible can observe the connected state. LCQ is inspired by the policy in [
In this paper, we establish the delay optimality of a two-hop network model with time-varying connectivity. We show that JSQ-LCQ policy is strongly optimal, such that it minimizes the number of customers in the system in the stochastic ordering sense. We use the coupling argument to show that the queue length processes under JSQ-LCQ is stochastically dominated by any other feasible policy. However, unlike typical coupling, we show that the coupled systems can exist in proposes JSQ-LCQ and proves the delay optimality of the algorithm for two-hop relay networks with time-varying connectivity, which is of theoretical significance; introduces a novel coupling technique associated with the transition of system phases defined in terms of the majorization relations among source and relay queues.
This paper is organized as follows. We present related works in Section
Delay-optimal scheduling is not only important from an engineering perspective but also of theoretical and mathematical interest. Delay optimality is notoriously hard to achieve with time-varying service capacity, and there exist only a few results which we review below. In their seminal work [
Our scheme can be regarded as a relay selection and scheduling scheme for two-hop cooperative relay networks, that is, a cooperative transmission utilizing multiple RNs, for example, [
In contrast to delay optimality,
Consider a time-slotted system consisting of one SQ and
The number of customers at SQ (resp., RQs) at time
Next we consider the condition for stability. The arrival rate to the system is
In this section, we will prove the delay optimality of JSQ-LCQ. We will show that JSQ-LCQ is optimal in the stochastic ordering sense which we define as follows. For two random variables
Let
Note that
Let
Theorem
Let
Unlike previous works on single-hop scheduling, the coupling argument in our problem must consider dynamic interactions among queues, connectivity, and the half-duplex constraint, as follows. JSQ-LCQ prioritizes serving the RQs; that is, it will serve the RQs whenever possible, in a balanced manner. Thus, the RQs will tend to be short under JSQ-LCQ. This means that, due to half-duplex operation, the SQ will get relatively long. However, if many RQs become empty due to prioritized service, the number of nonempty and connected RQs will become small. Hence JSQ-LCQ may be forced to frequently route customers to the RQs, in a balanced manner, in which case the RQs will build up. However, as the number of nonempty RQs grows, there will be many nonempty and connected RQs, and JSQ-LCQ will again begin to actively serve the RQs. Thus, we observe that the system exhibits some cyclic patterns in the services and evolution of the queue states.
Based on this observation, we identify that there exist
For vector
For two vectors
Recall that
We say the system is in weak majorization (WM) phase if the following relation holds: equivalently, there exists integer
We discuss the implication of the WM phase. Firstly, JSQ-LCQ will greedily serve the RQs whenever possible, making the overall length of RQs small. By contrast, the customers that arrived at the SQ will have to wait relatively long due to the half-duplex constraint. Condition (
Example of the system in weak majorization (WM) phase. The figure in the above (resp., below) shows the queues under policy JSQ-LCQ or
However, the system may get out of WM phase if most of the RQs are emptied out, after which JSQ-LCQ will mainly serve the SQ. Consequently, the RQs will become relatively long but the SQ will become relatively short, in contrast to WM phase. In that case, the system makes the transition to WFM phase, which we will define and discuss in detail in Section
To use forward induction we will show that if the system is in WM phase at time
Consider the queue length processes defined in Theorem
Initially, at
Now consider the system at time
For instance,
Suppose the service occurred at queue
Next we check if weak majorization (
Firstly consider the case where
Secondly, consider the case where
We will show that
Firstly, consider the case
Secondly, consider the case
Now suppose that the
Next we examine if (
We have the following definition.
Consider Let Let us add
We say that the system is in WFM phase if, at time
We discuss the implication of the WFM phase. As mentioned earlier, JSQ-LCQ prioritizes serving the RQs and hence will generate many empty RQs. As a result, JSQ-LCQ may be forced to route customers, resulting in a short SQ, for example, as in (
Example of the system in water-filling majorization (WFM) phase. The SQ under
In the proof of Lemma
Suppose there was a service from the Perform water-filling routing to
Let us denote the RQs after step (1) under
Next we consider the coupling of queues under
There exists a coupling between queue length processes
As previously, the queue connectivity is coupled such that
Secondly, suppose that the
Next we will show that Serve a customer from RQ Route Perform JSQ of a customer from
Let
Next, we consider step (3). Recall that the length of the
Perform water-filling routing of Perform JSQ from both
By induction hypothesis,
We are now ready to prove Theorem
Lemmas
One could ask, can we construct direct coupling between the processes of
In this section, we evaluate the performance of JSQ-LCQ via simulations. We used MATLAB as the discrete event simulator. A time-slotted system is simulated with the following parameters: the probability of a customer arriving at the source queue given by
Figure
Comparison of the average number of customers in the system versus the number of relay nodes.
Figure
Comparison of the average number of customers in the system versus the connectivity probability of relays.
Figure
Comparison of the average number of customers in the system versus the arrival rates.
In this paper, we studied a delay-optimal policy for half-duplex two-hop relay networks with symmetric connectivity on the service. We showed that JSQ-LCQ policy is strongly optimal; that is, the total queue length under JSQ-LCQ is stochastically dominated by any other policy. Our future work includes devising simple policies for two-hop relay networks with asymmetric connectivity and studying delay optimality for cooperative relay networks with more than two hops. Under certain realistic conditions for time-varying channels, for example, fading due to the user mobility having power-law distributions [
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported in part by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2016R1A2B1014934) and by Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (no. B0126-17-1046) and in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03930393).