LyX/Screencasts/queuing.lyx

   1 #LyX 1.5.3 created this file. For more info see http://www.lyx.org/\r
   2 \lyxformat 276\r
   3 \begin_document\r
   4 \begin_header\r
   5 \textclass article\r
   6 \language english\r
   7 \inputencoding auto\r
   8 \font_roman default\r
   9 \font_sans default\r
  10 \font_typewriter default\r
  11 \font_default_family default\r
  12 \font_sc false\r
  13 \font_osf false\r
  14 \font_sf_scale 100\r
  15 \font_tt_scale 100\r
  16 \graphics default\r
  17 \paperfontsize default\r
  18 \papersize default\r
  19 \use_geometry false\r
  20 \use_amsmath 1\r
  21 \use_esint 1\r
  22 \cite_engine basic\r
  23 \use_bibtopic false\r
  24 \paperorientation portrait\r
  25 \secnumdepth 3\r
  26 \tocdepth 3\r
  27 \paragraph_separation indent\r
  28 \defskip medskip\r
  29 \quotes_language english\r
  30 \papercolumns 1\r
  31 \papersides 1\r
  32 \paperpagestyle default\r
  33 \tracking_changes false\r
  34 \output_changes false\r
  35 \author "" \r
  36 \author "" \r
  37 \end_header\r
  38 \r
  39 \begin_body\r
  40 \r
  41 \begin_layout Standard\r
  42 \begin_inset Formula \[\r
  43 v_{k}=\begin{cases}\r
  44 k(b-k) & p=q\\\r
  45 \frac{b}{p-q}\frac{1-\left(\frac{q}{p}\right)^{k}}{1-\left(\frac{q}{p}\right)^{b}}-\frac{k}{p-q} & p\neq q\end{cases}\]\r
  46 \r
  47 \end_inset\r
  48 \r
  49 \r
  50 \newline\r
  51 \r
  52 \end_layout\r
  53 \r
  54 \begin_layout Standard\r
  55 \begin_inset Formula \[\r
  56 P^{n}=\left(\begin{array}{cc}\r
  57 1 & -\alpha\\\r
  58 1 & \beta\end{array}\right)\left(\begin{array}{cc}\r
  59 1 & 0\\\r
  60 0 & (1-\alpha-\beta)^{n}\end{array}\right)\left(\begin{array}{cc}\r
  61 1 & -\alpha\\\r
  62 1 & \beta\end{array}\right)\]\r
  63 \r
  64 \end_inset\r
  65 \r
  66 \r
  67 \newline\r
  68 \r
  69 \newline\r
  70 \r
  71 \newline\r
  72 \r
  73 \end_layout\r
  74 \r
  75 \begin_layout Standard\r
  76 \begin_inset Formula \[\r
  77 \sum_{j=0}^{\infty}\theta_{j}=\sum_{j=0}^{s-1}\frac{1}{j!}\left(\frac{\lambda}{\mu}\right)^{j}+\sum_{j=s}^{\infty}\frac{1}{s!}\left(\frac{\lambda}{\mu}\right)^{s}\left(\frac{\lambda}{s\mu}\right)^{j-s}\]\r
  78 \r
  79 \end_inset\r
  80 \r
  81 \r
  82 \end_layout\r
  83 \r
  84 \begin_layout Standard\r
  85 The traffic intensity in an \r
  86 \begin_inset Formula $M/M/s$\r
  87 \end_inset\r
  88 \r
  89  system is \r
  90 \begin_inset Formula $\rho=\lambda/(s\mu)$\r
  91 \end_inset\r
  92 \r
  93 .\r
  94  As the intensity of traffic approaches one, the mean queue length becomes\r
  95  unbounded.\r
  96  When \r
  97 \begin_inset Formula $\lambda<s\mu$\r
  98 \end_inset\r
  99 \r
 100 , then from (1.8) and (1.15), \r
 101 \begin_inset Formula \[\r
 102 \pi_{0}=\left\{ \sum_{j=0}^{s-1}\frac{1}{j!}\left(\frac{\lambda}{\mu}\right)^{j}+\frac{(\lambda/\mu)^{s}}{s!(1-\lambda/s\mu)}\right\} ^{-1}\]\r
 103 \r
 104 \end_inset\r
 105 \r
 106 and\r
 107 \newline\r
 108 \r
 109 \begin_inset Formula \[\r
 110 \pi_{k}=\begin{cases}\r
 111 \frac{1}{k!}\left(\frac{\lambda}{\mu}\right)^{k}\pi_{0} & \text{for\,}k=0,1,\ldots,s,\\\r
 112 \frac{1}{s!}\left(\frac{\lambda}{\mu}\right)^{s}\left(\frac{\lambda}{s\mu}\right)^{k-s}\pi_{0} & \text{for}\, k\geq s\end{cases}\]\r
 113 \r
 114 \end_inset\r
 115 \r
 116 \r
 117 \newline\r
 118 \r
 119 \newline\r
 120 \r
 121 \newline\r
 122 \r
 123 \begin_inset Float table\r
 124 wide false\r
 125 sideways false\r
 126 status open\r
 127 \r
 128 \begin_layout Standard\r
 129 \begin_inset Caption\r
 130 \r
 131 \begin_layout Standard\r
 132 Running times (seconds)\r
 133 \end_layout\r
 134 \r
 135 \end_inset\r
 136 \r
 137 \r
 138 \end_layout\r
 139 \r
 140 \begin_layout Standard\r
 141 \align center\r
 142 \begin_inset Tabular\r
 143 <lyxtabular version="3" rows="4" columns="4">\r
 144 <features>\r
 145 <column alignment="center" valignment="top" leftline="true" width="0">\r
 146 <column alignment="center" valignment="top" leftline="true" width="0">\r
 147 <column alignment="center" valignment="top" leftline="true" width="0">\r
 148 <column alignment="center" valignment="top" leftline="true" rightline="true" width="0">\r
 149 <row topline="true" bottomline="true">\r
 150 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 151 \begin_inset Text\r
 152 \r
 153 \begin_layout Standard\r
 154 Case\r
 155 \end_layout\r
 156 \r
 157 \end_inset\r
 158 </cell>\r
 159 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 160 \begin_inset Text\r
 161 \r
 162 \begin_layout Standard\r
 163 Method 1\r
 164 \end_layout\r
 165 \r
 166 \end_inset\r
 167 </cell>\r
 168 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 169 \begin_inset Text\r
 170 \r
 171 \begin_layout Standard\r
 172 Method 2\r
 173 \end_layout\r
 174 \r
 175 \end_inset\r
 176 </cell>\r
 177 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">\r
 178 \begin_inset Text\r
 179 \r
 180 \begin_layout Standard\r
 181 Method 3\r
 182 \end_layout\r
 183 \r
 184 \end_inset\r
 185 </cell>\r
 186 </row>\r
 187 <row topline="true">\r
 188 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 189 \begin_inset Text\r
 190 \r
 191 \begin_layout Standard\r
 192 1\r
 193 \end_layout\r
 194 \r
 195 \end_inset\r
 196 </cell>\r
 197 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 198 \begin_inset Text\r
 199 \r
 200 \begin_layout Standard\r
 201 23\r
 202 \end_layout\r
 203 \r
 204 \end_inset\r
 205 </cell>\r
 206 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 207 \begin_inset Text\r
 208 \r
 209 \begin_layout Standard\r
 210 47\r
 211 \end_layout\r
 212 \r
 213 \end_inset\r
 214 </cell>\r
 215 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">\r
 216 \begin_inset Text\r
 217 \r
 218 \begin_layout Standard\r
 219 389\r
 220 \end_layout\r
 221 \r
 222 \end_inset\r
 223 </cell>\r
 224 </row>\r
 225 <row topline="true">\r
 226 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 227 \begin_inset Text\r
 228 \r
 229 \begin_layout Standard\r
 230 2\r
 231 \end_layout\r
 232 \r
 233 \end_inset\r
 234 </cell>\r
 235 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 236 \begin_inset Text\r
 237 \r
 238 \begin_layout Standard\r
 239 43\r
 240 \end_layout\r
 241 \r
 242 \end_inset\r
 243 </cell>\r
 244 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 245 \begin_inset Text\r
 246 \r
 247 \begin_layout Standard\r
 248 53\r
 249 \end_layout\r
 250 \r
 251 \end_inset\r
 252 </cell>\r
 253 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">\r
 254 \begin_inset Text\r
 255 \r
 256 \begin_layout Standard\r
 257 393\r
 258 \end_layout\r
 259 \r
 260 \end_inset\r
 261 </cell>\r
 262 </row>\r
 263 <row topline="true" bottomline="true">\r
 264 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 265 \begin_inset Text\r
 266 \r
 267 \begin_layout Standard\r
 268 3\r
 269 \end_layout\r
 270 \r
 271 \end_inset\r
 272 </cell>\r
 273 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 274 \begin_inset Text\r
 275 \r
 276 \begin_layout Standard\r
 277 96\r
 278 \end_layout\r
 279 \r
 280 \end_inset\r
 281 </cell>\r
 282 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">\r
 283 \begin_inset Text\r
 284 \r
 285 \begin_layout Standard\r
 286 87\r
 287 \end_layout\r
 288 \r
 289 \end_inset\r
 290 </cell>\r
 291 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">\r
 292 \begin_inset Text\r
 293 \r
 294 \begin_layout Standard\r
 295 426\r
 296 \end_layout\r
 297 \r
 298 \end_inset\r
 299 </cell>\r
 300 </row>\r
 301 </lyxtabular>\r
 302 \r
 303 \end_inset\r
 304 \r
 305 \r
 306 \end_layout\r
 307 \r
 308 \begin_layout Standard\r
 309 \r
 310 \end_layout\r
 311 \r
 312 \end_inset\r
 313 \r
 314 \r
 315 \end_layout\r
 316 \r
 317 \end_body\r
 318 \end_document\r