第3章 愛(ài)爾蘭拒絕和等待系統(tǒng)

1、通信網(wǎng)基礎(chǔ)第三章 愛(ài)爾蘭拒絕與等待系統(tǒng)無(wú)線通信與網(wǎng)絡(luò)研究室李屹 博士/副教授/碩導(dǎo)2拒絕系統(tǒng)、等待系統(tǒng)o通信網(wǎng)絡(luò)中n信息速率可變；n信息流的不確定性，沒(méi)有規(guī)律；n大量終端的信息流有統(tǒng)計(jì)規(guī)律性。o實(shí)際系統(tǒng)中，呼叫遇到無(wú)可用資源時(shí)：3通信網(wǎng)與排隊(duì)論中術(shù)語(yǔ)對(duì)應(yīng)關(guān)系o電話網(wǎng)中基本指標(biāo)n中繼線，s 條n電話呼叫流的到達(dá)率 n一個(gè)呼叫，在中繼線有空閑時(shí)，占用中繼線，完成接續(xù)；n系統(tǒng)中s 條中繼線全部忙時(shí)，呼叫被拒絕。電話交換系統(tǒng)4業(yè)務(wù)量、呼叫量（話務(wù)量）5業(yè)務(wù)量、呼叫量（話務(wù)量）6業(yè)務(wù)量、呼叫量（話務(wù)量）7愛(ài)爾蘭系統(tǒng)、恩格謝特系統(tǒng)8時(shí)間阻塞率、呼叫阻塞率（呼損）9時(shí)延的分析10網(wǎng)絡(luò)分析方法網(wǎng)絡(luò)的平均呼損和

2、平均時(shí)延，是評(píng)價(jià)網(wǎng)絡(luò)性能的重要指標(biāo)，為網(wǎng)絡(luò)規(guī)劃和優(yōu)化建立基礎(chǔ)。11網(wǎng)絡(luò)分析方法在電話網(wǎng)中，僅需要描述兩個(gè)端之間的呼叫量，不需要區(qū)別方向；但是數(shù)據(jù)網(wǎng)絡(luò)中，需要在計(jì)算時(shí)區(qū)分端點(diǎn)的方向。12網(wǎng)絡(luò)分析方法13愛(ài)爾蘭即時(shí)拒絕系統(tǒng)-M/M/s(s)14狀態(tài)轉(zhuǎn)移圖o用系統(tǒng)中的呼叫數(shù)表示狀態(tài)，這個(gè)排隊(duì)系統(tǒng)是一個(gè)生滅過(guò)程o的達(dá)到率和離去率分別為：15穩(wěn)態(tài)分析o根據(jù)生滅過(guò)程的穩(wěn)態(tài)分布o(jì)令 ，并根據(jù)概率歸一性o解得o從而穩(wěn)態(tài)分布為：16愛(ài)爾蘭B公式17 愛(ài)爾蘭-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中繼數(shù)中繼數(shù)s呼叫量呼叫量a18全利用度系統(tǒng)、部分利用度系統(tǒng)19例1：M/M/系統(tǒng)的平均隊(duì)長(zhǎng)20例1：M/M/系統(tǒng)的平均隊(duì)長(zhǎng)上式中的pk 服從參數(shù)為a 的Poisson 分布，如果N為系統(tǒng)中的呼叫數(shù)，則21M/M/與M/M/s(s)的區(qū)別22例2：

10、M/M/s(s)系統(tǒng)的通過(guò)呼叫量23例2：M/M/s(s)系統(tǒng)的通過(guò)呼叫量24例3：大群化效應(yīng)o 一般來(lái)說(shuō)，社會(huì)服務(wù)資源在一定范圍內(nèi)統(tǒng)一利用要優(yōu)于分散經(jīng)營(yíng)，通信網(wǎng)中的信道資源也有類似的規(guī)律。o在保障一定通信質(zhì)量指標(biāo)的前提下，變分散利用的信道為集中利用的信道，有效提高網(wǎng)絡(luò)效率，這就是所謂通信線路大群化。25例3：大群化效應(yīng)26例3：大群化效應(yīng) 這種集中也有負(fù)面影響，因?yàn)楹艚辛靠赡軙?huì)波動(dòng)，在同樣的波動(dòng)水平下，大容量的中繼線群上的呼損將上升較多。27例3：大群化效應(yīng)28 愛(ài)爾蘭-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中繼數(shù)中繼數(shù)s呼叫量呼叫量a2930補(bǔ)充：綜合效應(yīng)31例4：中繼線順序限制32例5：主備線即時(shí)拒絕系統(tǒng)33例5：主備線即時(shí)拒絕系統(tǒng)34例5：主備線即時(shí)拒絕系統(tǒng)35愛(ài)爾蘭等待制系統(tǒng)36愛(ài)爾蘭等待制系統(tǒng)37愛(ài)爾蘭等待制

18、系統(tǒng)狀態(tài)轉(zhuǎn)移圖oM/M/S狀態(tài)轉(zhuǎn)移圖38愛(ài)爾蘭等待制系統(tǒng)狀態(tài)轉(zhuǎn)移圖oM/M/S狀態(tài)轉(zhuǎn)移圖sk sk 39愛(ài)爾蘭等待制系統(tǒng)系統(tǒng)分析n系統(tǒng)是一個(gè)n該各個(gè)狀態(tài)的和如下：k2 , 1 , 0kskskskk1, 2 , 1 n假設(shè) 為穩(wěn)態(tài)分布， 為平均數(shù)目，則： kpaskpssaskpkapskkkk00!0!根據(jù)Little定理，平均延遲為：NT 40愛(ài)爾蘭等待制系統(tǒng)系統(tǒng)分析 n根據(jù)概率歸一性， ，則：10kkpsksksskksasakap!1100 n在 的條件下，該系統(tǒng)有，且n以上給出了系統(tǒng)的穩(wěn)態(tài)分布sasasakapsskk11!110041愛(ài)爾蘭等待制系統(tǒng)系統(tǒng)分析o愛(ài)爾蘭 公式（Erla

19、ng C）n用來(lái)計(jì)算一個(gè)呼叫等待的概率n計(jì)算概率 ， 為需要等待的時(shí)間n呼叫到達(dá)系統(tǒng)的瞬間，不算該呼叫系統(tǒng)狀態(tài)分布為n一般 與 不同，如果到達(dá)的呼叫流為泊松過(guò)程，則：0wpw k k kp2 , 1kpkk42愛(ài)爾蘭等待制系統(tǒng)系統(tǒng)分析n一個(gè)呼叫到來(lái)且系統(tǒng)狀態(tài)處于 時(shí)，呼叫需要等待，需要等待的概率計(jì)算如下：skksasapsasapsspwpsksksskkskk1!000n一般被記為：n在 的條件下，系統(tǒng)有穩(wěn)態(tài)。由于該系統(tǒng)不拒絕呼叫，通過(guò)的呼叫量應(yīng)該為sapsaasCs1!,0asa43愛(ài)爾蘭B VS. 愛(ài)爾蘭C44Erlang B &C的簡(jiǎn)易遞歸計(jì)算公式o習(xí)題3.1o習(xí)題3.244(1, )( , )(1, )aB saB s asaB sa( , )( , ),1( , )sB s aC s asasaB