document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193728>Question 1193728</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The are two postal workers serving clients at the same time at the post office. Service time per client for the first postal worker is X1∼ E(λ1=0.94), second one - X2∼ E(λ2=1.0). When two new clients A and B walk into the postal office both workers are busy with a client already. Both clients want to be served by different workers. How much time on average would both clients A and B need to wait to be served, if A should be served first? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848600 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848600\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's calculate the average waiting time for clients A and B.\r<BR>\n" );
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document.write( "**1. Waiting Time for Client A**\r<BR>\n" );
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document.write( "* Client A waits for the first worker to finish serving the current client. <BR>\n" );
document.write( "* Since the service time of the first worker follows an exponential distribution with rate λ1 = 0.94, the waiting time for client A is also exponentially distributed with the same rate.\r<BR>\n" );
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document.write( "* **Average waiting time for Client A:** <BR>\n" );
document.write( "   E[Waiting Time for A] = 1 / λ1 = 1 / 0.94 ≈ 1.0638 \r<BR>\n" );
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document.write( "**2. Waiting Time for Client B**\r<BR>\n" );
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document.write( "* Client B waits for two events to occur:<BR>\n" );
document.write( "    * The first worker finishes serving client A. <BR>\n" );
document.write( "    * The second worker finishes serving the current client.\r<BR>\n" );
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document.write( "* The waiting time for Client B follows a minimum distribution of two independent exponential random variables. \r<BR>\n" );
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document.write( "* **Minimum of two exponential distributions:** <BR>\n" );
document.write( "   If X1 ~ Exp(λ1) and X2 ~ Exp(λ2), then the minimum of X1 and X2 follows an exponential distribution with rate (λ1 + λ2).\r<BR>\n" );
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document.write( "* **Average waiting time for Client B:**<BR>\n" );
document.write( "   E[Waiting Time for B] = 1 / (λ1 + λ2) = 1 / (0.94 + 1.0) ≈ 0.5263\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **Average waiting time for Client A:** Approximately 1.0638 units of time<BR>\n" );
document.write( "* **Average waiting time for Client B:** Approximately 0.5263 units of time\r<BR>\n" );
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document.write( "**Note:** <BR>\n" );
document.write( "* This analysis assumes that the service times for both workers are independent and identically distributed.<BR>\n" );
document.write( "* The units of time will depend on the units used for the service rates (e.g., minutes, seconds).\r<BR>\n" );
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document.write( "Let me know if you have any other questions or would like to explore different scenarios!<BR>\n" );
document.write( " </SPAN>\n" );
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