A Tandem <svg style="vertical-align:-2.934pt;width:240.825px;" id="M1" height="19.5375" version="1.1" viewBox="0 0 240.825 19.5375" width="240.825" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,15.825)"><path id="x42" 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" /></g><g transform="matrix(.022,-0,0,-.022,13.243,15.825)"><path id="x4D" 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" /></g><g transform="matrix(.022,-0,0,-.022,33.216,15.825)"><path id="x41" 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" /></g><g transform="matrix(.022,-0,0,-.022,48.706,15.825)"><path id="x50" 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" /></g><g transform="matrix(.022,-0,0,-.022,61.326,15.825)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.022,-0,0,-.022,70.562,15.825)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242
q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g><g transform="matrix(.022,-0,0,-.022,86.791,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,96.026,15.825)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.022,-0,0,-.022,116.739,15.825)"><path id="x2192" d="M901 255q-71 -62 -185 -187l-22 15l102 147h-727v50h727l-102 147l22 15q114 -125 185 -187z" /></g><g transform="matrix(.022,-0,0,-.022,148.077,15.825)"><path id="x2022" d="M455 252q0 -79 -57 -136t-136 -57q-80 0 -136 56q-56 58 -56 137q0 77 56 135q57 57 136 57q84 0 138 -56q55 -55 55 -136z" /></g><g transform="matrix(.022,-0,0,-.022,159.801,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,169.036,15.825)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g><g transform="matrix(.022,-0,0,-.022,191.923,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,201.159,15.825)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g><g transform="matrix(.022,-0,0,-.022,220.75,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,229.986,15.825)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g></svg> Queue with Group Occupation of Servers at the Second Station

<table>The objective function as a function of the number of servers in Station 2 for different service rates.</table>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: A Tandem <svg style="vertical-align:-2.934pt;width:240.825px;" id="M1" height="19.5375" version="1.1" viewBox="0 0 240.825 19.5375" width="240.825" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,15.825)"><path id="x42" 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" /></g><g transform="matrix(.022,-0,0,-.022,13.243,15.825)"><path id="x4D" 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" /></g><g transform="matrix(.022,-0,0,-.022,33.216,15.825)"><path id="x41" 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" /></g><g transform="matrix(.022,-0,0,-.022,48.706,15.825)"><path id="x50" 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" /></g><g transform="matrix(.022,-0,0,-.022,61.326,15.825)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.022,-0,0,-.022,70.562,15.825)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242
q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g><g transform="matrix(.022,-0,0,-.022,86.791,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,96.026,15.825)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.022,-0,0,-.022,116.739,15.825)"><path id="x2192" d="M901 255q-71 -62 -185 -187l-22 15l102 147h-727v50h727l-102 147l22 15q114 -125 185 -187z" /></g><g transform="matrix(.022,-0,0,-.022,148.077,15.825)"><path id="x2022" d="M455 252q0 -79 -57 -136t-136 -57q-80 0 -136 56q-56 58 -56 137q0 77 56 135q57 57 136 57q84 0 138 -56q55 -55 55 -136z" /></g><g transform="matrix(.022,-0,0,-.022,159.801,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,169.036,15.825)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g><g transform="matrix(.022,-0,0,-.022,191.923,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,201.159,15.825)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g><g transform="matrix(.022,-0,0,-.022,220.75,15.825)"><use xlink:href="#x2F" /></g><g transform="matrix(.022,-0,0,-.022,229.986,15.825)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g></svg> Queue with Group Occupation of Servers at the Second Station 

Mathematical Problems in Engineering

A Tandem Queue with Group Occupation of Servers at the Second Station

Figure 3