document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198576>Question 1198576</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During the registration period, St. Mary’s Univeristy has a technician in its service center to answer <BR>\n" );
document.write( "students’ questions. The number of telephone calls arriving at this center follows a Poisson <BR>\n" );
document.write( "distribution with an approximate average rate of 10/h. The time required to answer one call follows an <BR>\n" );
document.write( "exponential distribution with an average of 4 min. Answer the following questions: <BR>\n" );
document.write( "a. What is the average time between incoming calls? <BR>\n" );
document.write( "b. What is the average number of calls that the technician can attend in 1h? <BR>\n" );
document.write( "c. What is the probability of there being exactly four calls on hold at a given time? <BR>\n" );
document.write( "d. What is the probability of the number of calls in the system exceeding 10? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848272 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848272\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Average Time Between Incoming Calls**\r<BR>\n" );
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document.write( "* **Arrival Rate (λ):** 10 calls/hour <BR>\n" );
document.write( "* **Average Time Between Arrivals:** 1 / λ = 1 / 10 calls/hour = 0.1 hours/call <BR>\n" );
document.write( "* **Convert to minutes:** 0.1 hours/call * 60 minutes/hour = 6 minutes/call\r<BR>\n" );
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document.write( "**Therefore, the average time between incoming calls is 6 minutes.**\r<BR>\n" );
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document.write( "**b. Average Number of Calls Attended in 1 Hour**\r<BR>\n" );
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document.write( "* **Service Time per Call:** 4 minutes<BR>\n" );
document.write( "* **Calls Attended per Hour:** 60 minutes/hour / 4 minutes/call = 15 calls/hour\r<BR>\n" );
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document.write( "**Therefore, the technician can attend an average of 15 calls per hour.**\r<BR>\n" );
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document.write( "**c. Probability of Exactly Four Calls on Hold**\r<BR>\n" );
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document.write( "* To calculate this, we need to use queuing theory. <BR>\n" );
document.write( "* **Assumptions:**<BR>\n" );
document.write( "    * This is an M/M/1 queue (single-server queue with Poisson arrivals and exponential service times).<BR>\n" );
document.write( "    * The arrival rate (λ) is 10 calls/hour.<BR>\n" );
document.write( "    * The service rate (μ) is 15 calls/hour.<BR>\n" );
document.write( "* **Traffic Intensity (ρ):** ρ = λ / μ = 10 calls/hour / 15 calls/hour = 2/3 <BR>\n" );
document.write( "* **Probability of n Calls in the System (P_n):**<BR>\n" );
document.write( "    * P_n = (1 - ρ) * ρ^n <BR>\n" );
document.write( "    * For exactly four calls on hold (n = 4): <BR>\n" );
document.write( "        * P_4 = (1 - 2/3) * (2/3)^4 = (1/3) * (16/81) = 16/243 <BR>\n" );
document.write( "        * P_4 ≈ 0.0658\r<BR>\n" );
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document.write( "**Therefore, the probability of there being exactly four calls on hold at a given time is approximately 0.0658.**\r<BR>\n" );
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document.write( "**d. Probability of More Than 10 Calls in the System**\r<BR>\n" );
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document.write( "* To calculate this, we need to use the formula for the probability of n calls in the system for an M/M/1 queue:<BR>\n" );
document.write( "    * P_n = (1 - ρ) * ρ^n \r<BR>\n" );
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document.write( "* The probability of more than 10 calls in the system is:<BR>\n" );
document.write( "    * P(n > 10) = 1 - P(n ≤ 10) <BR>\n" );
document.write( "    * P(n > 10) = 1 - (P_0 + P_1 + P_2 + ... + P_10)\r<BR>\n" );
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document.write( "* Calculate P_n for n = 0 to 10 using the formula above. <BR>\n" );
document.write( "* Sum these probabilities to find P(n ≤ 10).<BR>\n" );
document.write( "* Subtract P(n ≤ 10) from 1 to find P(n > 10).\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* Calculating P(n > 10) manually can be tedious. You can use software tools like Excel, R, or specialized queuing theory software to calculate this probability more efficiently.\r<BR>\n" );
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document.write( "This analysis provides a basic framework for understanding the call center's performance. \r<BR>\n" );
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document.write( "**Disclaimer:** <BR>\n" );
document.write( "* This analysis assumes a steady-state condition, where the arrival rate and service rate remain constant over time. <BR>\n" );
document.write( "* Real-world scenarios may involve variations in arrival rates and service times.<BR>\n" );
document.write( "* This analysis does not consider factors such as call abandonment, priority queues, or multiple servers.\r<BR>\n" );
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document.write( "I hope this helps! Let me know if you have any further questions.<BR>\n" );
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