在Apache Flink中，可以通过设置slot.timeout属性来解决任务管理器空闲超时问题。以下是一个示例代码，说明如何设置slot.timeout属性：

import org.apache.flink.configuration.Configuration;
import org.apache.flink.configuration.JobManagerOptions;
import org.apache.flink.runtime.jobmaster.JobMaster;
import org.apache.flink.runtime.jobmaster.slotpool.SlotPool;
import org.apache.flink.runtime.jobmaster.slotpool.SlotReport;
import org.apache.flink.runtime.jobmaster.slotpool.SlotRequest;
import org.apache.flink.runtime.resourcemanager.ResourceManager;
import org.apache.flink.runtime.resourcemanager.SlotRequestTimeoutListener;
import org.apache.flink.runtime.resourcemanager.slotmanager.SlotManager;
import org.apache.flink.runtime.taskexecutor.slot.SlotOffer;
import org.apache.flink.util.Preconditions;

import javax.annotation.Nullable;

public class CustomResourceManager extends ResourceManager {

    // 定义空闲超时时间
    private final long idleTimeout;

    public CustomResourceManager(
            Configuration configuration,
            SlotManager slotManager,
            SlotPool slotPool,
            JobMaster jobMaster,
            long idleTimeout) {
        super(configuration, slotManager, slotPool, jobMaster);

        Preconditions.checkArgument(idleTimeout >= 0, "Idle timeout must be >= 0");
        this.idleTimeout = idleTimeout;
    }

    @Override
    public void start() throws Exception {
        super.start();

        // 注册空闲超时监听器
        getFencingToken().getLeaderElectionService().addListener(
                new SlotRequestTimeoutListener(getFencingToken().toUUID(), getIdleTimeout()));
    }

    public long getIdleTimeout() {
        return idleTimeout;
    }

    public static CustomResourceManager create(
            Configuration configuration,
            SlotManager slotManager,
            SlotPool slotPool,
            JobMaster jobMaster,
            long idleTimeout) throws Exception {
        return new CustomResourceManager(configuration, slotManager, slotPool, jobMaster, idleTimeout);
    }

    public static void main(String[] args) throws Exception {
        Configuration configuration = new Configuration();
        // 设置任务管理器空闲超时时间为10分钟
        configuration.setLong(JobManagerOptions.SLOT_IDLE_TIMEOUT, 600000);

        // 创建自定义资源管理器
        CustomResourceManager resourceManager = create(
                configuration,
                slotManager,
                slotPool,
                jobMaster,
                600000);

        // 启动资源管理器
        resourceManager.start();
    }
}

在上述代码中，CustomResourceManager类继承自ResourceManager类，并添加了一个idleTimeout成员变量，表示任务管理器空闲超时时间。在start()方法中，注册了一个空闲超时监听器，当任务管理器空闲超时时，会触发该监听器。

在main()方法中，通过JobManagerOptions.SLOT_IDLE_TIMEOUT属性设置了任务管理器的空闲超时时间为10分钟（600000毫秒），然后创建了一个CustomResourceManager实例，并调用start()方法启动资源管理器。

请根据你的实际情况调整超时时间和其他配置。