Transient and Stationary Losses in a Finite-Buffer Queue with Batch Arrivals

<table>The autocorrelation at lag <i>k</i> of the sequence of interarrival times in <svg height="15.65" id="M157" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 52.912498 15.65" width="52.912498" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.362)"><path d="M372 352v-2q176 -27 176 -159q0 -99 -95 -151q-73 -40 -196 -40h-221v28q65 5 78.5 19t13.5 76v404q0 63 -12 77t-72 18v28h257q102 0 150 -34q26 -18 41 -48.5t15 -65.5q0 -111 -135 -150zM213 362h47q80 0 117.5 32t37.5 96q0 56 -33 92t-100 36q-41 0 -56 -10
q-13 -8 -13 -50v-196zM213 328v-203q0 -54 17 -72.5t66 -18.5q67 1 110.5 37t43.5 110q0 147 -192 147h-45z" id="x42"></path></g><g transform="matrix(.017,-0,0,-.017,10.058,11.362)"><path d="M861 0h-263v28q61 5 73 22t12 97l-4 408h-4l-244 -548h-20l-224 534h-2l-15 -285q-5 -98 -3 -166q1 -34 19.5 -45t72.5 -17v-28h-233v28q45 5 62.5 17t22.5 43q11 65 19 183l14 208q3 36 3.5 55t-2.5 37.5t-6.5 25t-16.5 13t-25.5 8t-39.5 4.5v28h164l228 -501l232 501
h170v-28q-68 -6 -81 -21.5t-11 -91.5l9 -362q2 -80 15 -97t78 -22v-28z" id="x4D"></path></g><g transform="matrix(.017,-0,0,-.017,25.204,11.362)"><path d="M673 0h-245v28q47 6 58 16t1 39l-55 154h-213q-31 -87 -36 -129q-16 -47 -5 -61t68 -19v-28h-226v28q50 7 68.5 23.5t43.5 80.5l214 524l24 9l193 -536q23 -63 41.5 -79.5t68.5 -21.5v-28zM418 280l-81 255h-2q-74 -197 -101 -255h184z" id="x41"></path></g><g transform="matrix(.017,-0,0,-.017,36.95,11.362)"><path d="M46 650h247q112 0 169 -41q64 -49 64 -141q0 -91 -56.5 -143t-138.5 -62q-23 -3 -45 -1l-74 19v-157q0 -63 15.5 -77t83.5 -19v-28h-271v28q60 5 74 20t14 77v400q0 65 -12.5 79t-69.5 18v28zM212 559v-245q31 -12 73 -12q59 0 102 38t43 124q0 77 -42.5 115.5
t-108.5 38.5q-44 0 -56 -12q-11 -8 -11 -47z" id="x50"></path></g>
<g transform="matrix(.012,-0,0,-.012,46.525,15.45)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g>
</svg>.</table>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Transient and Stationary Losses in a Finite-Buffer Queue with Batch Arrivals 