QUEUEING ANALYSIS$$

All customer inquiries will be handled by the call center.$$

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Given the demographics of the US population, your client must provide services in both English and Spanish.$$ There are two options:

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Configuration$-1$: This configuration retains separate queues for each language. The customers will dial a toll$-$free number to reach a contact center. The phone system will ask each caller to choose English or Spanish as their preferred language as soon as the call is received. The call is then routed to the queue served by the Spanish$-$speaking customer service representatives or to the queue that is served by English$-$speaking customer service representatives. From a customer support perspective, the customer service representatives will be $$monolingual$,$ i$.$e$.,$ each employee is fluent in their own spoken language and have been trained to respond to calls in that language only. These representatives are not fluent enough to assist the customers with their inquiries in a second language.

In Configuration$-1,$ there are practically two separate queueing systems:$$ one for English$-$speaking callers and one for Spanish$-$speaking callers. Each queue receives the calls by callers in one language and the calls are served by any of the available monolingual customer service representatives.$$

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Configuration$-2$: In this configuration, there will be only one queue, served by bilingual customer service represetatives. This requires that your client hire bilingual employees that are fluent in both English and Spanish. The contact center will$$ ask a caller to choose their preferred language, but all incoming calls will join the same queue only to be greeted in their chosen language by a customer service representative when the customer service representative is ready to serve them.

In Configuration$-2,$ There is one queueing system: calls join one queue and get served by any of the bilingual customer service representatives.

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You should use the following information in your analysis:$$

$$ The customer service times are variable with CV$=1\mathrm{.}5.$ This is true regardless of the type of call, the language, and the skill level of the customer service representatives.

$$ The time between arrivals is variable and not predictable. The number of calls received is Poisson distributed $($hence CV$=1).$ This is true regardless of the type of call, the language, and the skill level of the customer service representatives.

$$ A monolingual customer service representative responds to calls only in their spoken language.

$$ Calls waiting in a queue will remain in the same queue until the time of service $($i$.$e$.$ no reneging, no balking$).$

$$ The contact center has ample $$virtual capacity$($i$.$e$.$ there is no limit on the queue length$).$

$$ All customers are served first$-$come$-$first$-$served $($FCFS$)$ once they join a queue.

$$ Each customer is expected to be served in $8$ minutes on the average.

$$ The average call volume of Spanish$-$speaking customers varies by day of the week and by hour of day; the average Spanish$-$speaking call volume can be as high as $12$ calls per hour at any time. The average call volume of English$-$speaking customers varies by day of the week and by hour of day; the average English$-$speaking call volume can be as high as $20$ calls per hour at any time.The total call volume $($the total of Spanish$-$ and English$-$speaking calls received$)$ also varies by hour of day and day of week; the total call volume can be as high as $20$ calls per hour on the average at any time.

$$ The call center will operate $24$ hours a day, covered by three $8-$hour shifts.

$$ Your client has set the target performance at the call center: The caller's average wait time in the queue should not exceed $3$ minutes. $$The client has limited budget and wants to know the minimum number of customer service representatives needed to meet this target.

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A$)$ The effect of variability:

Suppose the statistical distribution of time between arrivals and service times have changed. There is more variability in interarrival times and the time required to complete service for each caller.$$ The coefficient of variation of both time between arrivals are the service times have gone up to $2\mathrm{.}5.$

i$.(5$ points$)$ Complete Table$-$Q$1-3$E$,$ Table$-$Q$1-3$S$,$ and Table$-$Q$1-4$ based on the new information.