第3章 爱尔兰拒绝和等待系统

1、通信网基础第三章 爱尔兰拒绝与等待系统无线通信与网络研究室李屹 博士/副教授/硕导2拒绝系统、等待系统o通信网络中n信息速率可变；n信息流的不确定性，没有规律；n大量终端的信息流有统计规律性。o实际系统中，呼叫遇到无可用资源时：3通信网与排队论中术语对应关系o电话网中基本指标n中继线，s 条n电话呼叫流的到达率 n一个呼叫，在中继线有空闲时，占用中继线，完成接续；n系统中s 条中继线全部忙时，呼叫被拒绝。电话交换系统4业务量、呼叫量（话务量）5业务量、呼叫量（话务量）6业务量、呼叫量（话务量）7爱尔兰系统、恩格谢特系统8时间阻塞率、呼叫阻塞率（呼损）9时延的分析10网络分析方法网络的平均呼损和

2、平均时延，是评价网络性能的重要指标，为网络规划和优化建立基础。11网络分析方法在电话网中，仅需要描述两个端之间的呼叫量，不需要区别方向；但是数据网络中，需要在计算时区分端点的方向。12网络分析方法13爱尔兰即时拒绝系统-M/M/s(s)14状态转移图o用系统中的呼叫数表示状态，这个排队系统是一个生灭过程o的达到率和离去率分别为：15稳态分析o根据生灭过程的稳态分布o令 ，并根据概率归一性o解得o从而稳态分布为：16爱尔兰B公式17 爱尔兰-B表sBsB0.01%0.01%0.02%0.02%0.05%0.05%0.1%0.1%0.2%0.2%0.5%0.5%1.0%1.0%2.0%2.0%5.

3、0%5.0%10%10%20%20%50%50%1 10.00010.00010.00020.00020.00050.00050.00100.00100.00200.00200.00500.00500.01010.01010.02040.02040.05260.05260.1110.1110.2500.2501.001.002 20.01420.01420.02020.02020.03210.03210.04580.04580.06530.06530.1050.1050.1530.1530.2230.2230.3810.3810.5950.5951.001.002.732.733 30.086

4、80.08680.1100.1100.1520.1520.1940.1940.2490.2490.3490.3490.4550.4550.6050.6050.8990.8991.271.271.931.934.594.594 40.2350.2350.2820.2820.3620.3620.4390.4390.5350.5350.7010.7010.8690.8691.091.091.521.522.052.052.952.956.506.50. . . . . . . . . . . . . .10102.262.262.472.472.802.803.093.093.423.423.963

5、.964.464.465.085.086.226.227.517.519.699.6918.318.3. . . . . . . . . . . . . .20207.707.708.158.158.838.839.419.4110.110.111.111.112.012.013.213.215.315.317.617.621.621.638.238.2303014.214.214.914.915.915.916.716.717.617.619.019.020.320.321.921.924.824.828.128.133.833.858.158.1404021.421.422.222.223

6、.423.424.424.425.625.627.427.429.029.031.031.034.634.638.838.846.146.178.178.1505028.928.929.829.831.331.332.532.533.933.936.036.037.937.940.340.344.544.549.649.658.558.598.198.110010069.369.370.970.973.273.275.275.277.577.580.980.984.184.188.088.095.295.2104.1104.1120.6120.6198.0198.0200200156.2156

7、.2158.7158.7162.5162.5165.6165.6169.2169.2174.6174.6179.7179.7186.2186.2198.5198.5214.3214.3245.4245.4398.0398.0300300246.4246.4249.7249.7254.6254.6258.6258.6263.2263.2270.4270.4277.1277.1285.7285.7302.6302.6325.0325.0370.3370.3598.0598.0500500431.4431.4435.9435.9442.5442.5448.2448.2454.5454.5464.54

8、64.5474.0474.0486.4486.4511.8511.8546.7546.7620.2620.2998.0998.0800800714.3714.3720.3720.3729.1729.1736.6736.6745.1745.1758.7758.7771.8771.8789.3789.3826.4826.4879.7879.7995.1995.11598.01598.0900900809.4809.4815.8815.8825.3825.3833.3833.3842.5842.5857.2857.2871.5871.5890.5890.5931.4931.4990.8990.811

9、20.11120.11798.01798.010001000904.8904.8911.7911.7921.7921.7930.3930.3940.1940.1955.9955.9971.2971.2991.9991.91036.41036.41101.81101.81245.11245.11998.01998.0aasBps，),(阻塞率阻塞率ps中继数中继数s呼叫量呼叫量a18全利用度系统、部分利用度系统19例1：M/M/系统的平均队长20例1：M/M/系统的平均队长上式中的pk 服从参数为a 的Poisson 分布，如果N为系统中的呼叫数，则21M/M/与M/M/s(s)的区别22例2：

10、M/M/s(s)系统的通过呼叫量23例2：M/M/s(s)系统的通过呼叫量24例3：大群化效应o 一般来说，社会服务资源在一定范围内统一利用要优于分散经营，通信网中的信道资源也有类似的规律。o在保障一定通信质量指标的前提下，变分散利用的信道为集中利用的信道，有效提高网络效率，这就是所谓通信线路大群化。25例3：大群化效应26例3：大群化效应 这种集中也有负面影响，因为呼叫量可能会波动，在同样的波动水平下，大容量的中继线群上的呼损将上升较多。27例3：大群化效应28 爱尔兰-B表sBsB0.01%0.01%0.02%0.02%0.05%0.05%0.1%0.1%0.2%0.2%0.5%0.5%1

11、.0%1.0%2.0%2.0%5.0%5.0%10%10%20%20%50%50%1 10.00010.00010.00020.00020.00050.00050.00100.00100.00200.00200.00500.00500.01010.01010.02040.02040.05260.05260.1110.1110.2500.2501.001.002 20.01420.01420.02020.02020.03210.03210.04580.04580.06530.06530.1050.1050.1530.1530.2230.2230.3810.3810.5950.5951.001.0

12、02.732.733 30.08680.08680.1100.1100.1520.1520.1940.1940.2490.2490.3490.3490.4550.4550.6050.6050.8990.8991.271.271.931.934.594.594 40.2350.2350.2820.2820.3620.3620.4390.4390.5350.5350.7010.7010.8690.8691.091.091.521.522.052.052.952.956.506.50. . . . . . . . . . . . . .10102.262.262.472.472.802.803.09

13、3.093.423.423.963.964.464.465.085.086.226.227.517.519.699.6918.318.3. . . . . . . . . . . . . .20207.707.708.158.158.838.839.419.4110.110.111.111.112.012.013.213.215.315.317.617.621.621.638.238.2303014.214.214.914.915.915.916.716.717.617.619.019.020.320.321.921.924.824.828.128.133.833.858.158.140402

14、1.421.422.222.223.423.424.424.425.625.627.427.429.029.031.031.034.634.638.838.846.146.178.178.1505028.928.929.829.831.331.332.532.533.933.936.036.037.937.940.340.344.544.549.649.658.558.598.198.110010069.369.370.970.973.273.275.275.277.577.580.980.984.184.188.088.095.295.2104.1104.1120.6120.6198.019

15、8.0200200156.2156.2158.7158.7162.5162.5165.6165.6169.2169.2174.6174.6179.7179.7186.2186.2198.5198.5214.3214.3245.4245.4398.0398.0300300246.4246.4249.7249.7254.6254.6258.6258.6263.2263.2270.4270.4277.1277.1285.7285.7302.6302.6325.0325.0370.3370.3598.0598.0500500431.4431.4435.9435.9442.5442.5448.2448.

16、2454.5454.5464.5464.5474.0474.0486.4486.4511.8511.8546.7546.7620.2620.2998.0998.0800800714.3714.3720.3720.3729.1729.1736.6736.6745.1745.1758.7758.7771.8771.8789.3789.3826.4826.4879.7879.7995.1995.11598.01598.0900900809.4809.4815.8815.8825.3825.3833.3833.3842.5842.5857.2857.2871.5871.5890.5890.5931.4

17、931.4990.8990.81120.11120.11798.01798.010001000904.8904.8911.7911.7921.7921.7930.3930.3940.1940.1955.9955.9971.2971.2991.9991.91036.41036.41101.81101.81245.11245.11998.01998.0aasBps，),(阻塞率阻塞率ps中继数中继数s呼叫量呼叫量a2930补充：综合效应31例4：中继线顺序限制32例5：主备线即时拒绝系统33例5：主备线即时拒绝系统34例5：主备线即时拒绝系统35爱尔兰等待制系统36爱尔兰等待制系统37爱尔兰等待制

18、系统状态转移图oM/M/S状态转移图38爱尔兰等待制系统状态转移图oM/M/S状态转移图sk sk 39爱尔兰等待制系统系统分析n系统是一个n该各个状态的和如下：k2 , 1 , 0kskskskk1, 2 , 1 n假设 为稳态分布， 为平均数目，则： kpaskpssaskpkapskkkk00!0!根据Little定理，平均延迟为：NT 40爱尔兰等待制系统系统分析 n根据概率归一性， ，则：10kkpsksksskksasakap!1100 n在 的条件下，该系统有，且n以上给出了系统的稳态分布sasasakapsskk11!110041爱尔兰等待制系统系统分析o爱尔兰 公式（Erla

19、ng C）n用来计算一个呼叫等待的概率n计算概率 ， 为需要等待的时间n呼叫到达系统的瞬间，不算该呼叫系统状态分布为n一般 与 不同，如果到达的呼叫流为泊松过程，则：0wpw k k kp2 , 1kpkk42爱尔兰等待制系统系统分析n一个呼叫到来且系统状态处于 时，呼叫需要等待，需要等待的概率计算如下：skksasapsasapsspwpsksksskkskk1!000n一般被记为：n在 的条件下，系统有稳态。由于该系统不拒绝呼叫，通过的呼叫量应该为sapsaasCs1!,0asa43爱尔兰B VS. 爱尔兰C44Erlang B &C的简易递归计算公式o习题3.1o习题3.244(1, )( , )(1, )aB saB s asaB sa( , )( , ),1( , )sB s aC s asasaB