Avishai Mandelbaum, Technion Sergey Zeltyn, IBM Research Lab, Haifa Staffing Many-Server Queues with Impatient Customers: Constraint Satisfaction in Call Centers Avishai Mandelbaum, Technion Sergey Zeltyn, IBM Research Lab, Haifa To replace the title / subtitle with your own: Click on the title block -> select all the text by pressing Ctrl+A -> press Delete key -> type your own text 11/15/2018

Main operational regimes: ED, QED, ED+QED. Plan of Talk Paper setup. Motivation. Paper contribution. Related research. Single-interval constraint satisfaction. Main operational regimes: ED, QED, ED+QED. Numerical results and practical guidelines. Global constraint satisfaction. QED regime. ED regime. Possible future research. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Paper Setup M/M/n+G Queueing Model. Poisson arrival rate λ, assume (large arrival rate). Exponential service times with rate μ (fixed), E[S]=1/μ. n servers, according to staffing rule n=F(λ). General patience distribution G (fixed) with mean . Operational Performance Measures. Probability to abandon P{Ab}. Average wait E[W]. Delay probability P{W>0}. Probability to exceed a deadline P{W>T}, T>0. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Motivation: Constraint Satisfaction in Service Systems Service-Level Agreements. Prevalent in many applications: call centers, IT support, business process management. Examples: 80% of callers should be served within 30 seconds, fraction of abandoning customers should not exceed 3%. How to satisfy SLA and minimize staffing costs? Constraint satisfaction on a single interval. Find minimal staffing level that satisfies SLA for each basic interval (say, one hour). Mainstream theoretical approach. Global constraint satisfaction. Minimize staffing costs for multi-interval SLA (daily, weekly, monthly). Typical practical setup. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Paper Contribution Constraint satisfaction on a single interval Asymptotic results for three operational regimes derived: Efficiency-Driven (ED), Quality and Efficiency-Driven (QED), ED+QED – new. Practical recommendations on applications of the three regimes. Global constraint satisfaction. Global constraint problem studied in QED and ED regimes. Importance of monotone patience hazard rate assumption in the ED regime. Insights on differences between local and global solutions: based on theoretical results and numerical experiments. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Related Research Constraint satisfaction and cost minimization: Borst, Mandelbaum and Reiman, 2004. M/M/n+G queue: Baccelli and Hebuterne, 1981. Zeltyn and Mandelbaum, 2005. QED operational regime: Halfin and Whitt, 1981. Garnett, Mandelbaum and Reiman, 2002. ED+QED regime: Baron and Milner, 2008. Global constraint satisfaction: Koole and van der Sluis, 2003. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Asymptotic Operational Regimes Offered load: R=λ∙E[S], minutes of work that arrive per minute. Efficiency-Driven (ED): Quality and Efficiency-Driven (QED) regime: ED+QED regime: Asymptotic formulae for large λ, n are derived for all regimes. These formulae are used to solve single-interval constraint satisfaction problems. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Probability to Abandon: Approximations ED regime: QED regime: A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Single Interval Constraint Satisfaction: P{Abandon} Let μ=1 (time-units are minutes). ED approximations do not work well for small-to-moderate systems with strict constraints. Maybe QED approach always works well? That is not the case. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Single Interval Constraint Satisfaction: Average Wait ED regime: QED regime: ED approximations do not work well for small-to-moderate systems with strict constraints. QED does not work well for large systems with loose constraints, unless patience is exponential. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Single Interval Constraint Satisfaction: P{W>T} ED+QED regime: QED regime: A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Single-Interval Constraint Satisfaction: Conclusions Small-to-moderate systems, large systems with “strict constraints” Use QED regime/approximations. Large systems with “loose constraints” Probability to abandon: both QED and ED are good. Average wait: use ED. Tail probability P{W>T}: use ED+QED. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Global Constraint Satisfaction: General Setting Day of work consists of K basic intervals (assume same length). Poisson arrival rates per interval: Staffing costs per interval: ci, 1≤i≤K. Staffing vector: Optimal staffing level: with respect to global constraint. Asymptotically optimal staffing: Asymptotically feasible staffing: satisfies global constraint. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

QED Regime: Constraint on Probability to Abandon Global constraint: where overall probability to abandon Theorem. Introduce the optimization problem Then there exist at least one optimal solution of (1) Define Staffing is asymptotically feasible and optimal: A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

QED Regime: Discussion Conjectures for P{Ab} constraint (based on numerical study). Solution of optimization problem is unique. Assume equal costs at all intervals. Then optimization problem has a unique solution with 2nd conjecture implies higher service level for higher arrival rates. Convexity of P{Ab} in number of servers needed for the proof. Scaled constraint on average wait. QED staffing is asymptotically optimal. Constraint on delay probability. Either use QED staffing or “close gates” A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Probability to Abandon: Comparison between Global and Single-Interval Constraint Satisfaction Arrival rate based on data from Israeli telecom company. SLA’s: P{Ab}≤3%, P{W>0} ≤30%. 9 work-hours saved for first SLA, 17 work-hours for joint SLA (out of 3000). Global staffing does not add much for tight service constraints. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

ED Regime: Constraint on Average Wait Global constraint: where overall average wait Theorem. Define optimization problem where G is continuous and strictly decreasing over distribution support. Then there exists at least one optimal solution Define Staffing is asymptotically feasible and optimal: A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

ED Regime: Constraint on Average Wait – DHR case Equal staffing costs and G with strictly Decreasing Hazard Rate. Then the unique solution of (2) is where γ* solves Staffing for DHR case: Sustain same service level during all intervals. Explanation. Limit of average wait is convex in γ. Very low staffing levels lead to increasing cost. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

ED Regime: Constraint on Average Wait – IHR case Equal staffing costs and G with strictly Increasing Hazard Rate. Intervals with staffing around offered load: Intervals with essentially no staffing: Intervals with “intermediate” staffing: Then solution of (2) has the following properties: M2 consists of either zero or one element. Staffing for IHR case: differentiate service levels. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018

Possible Future Research Proving some conjectures for constraint satisfaction and cost optimization problems (convexity of probability to abandon in n needed for proofs). Multi-interval cost optimization. Generally distributed service times. Random arrival rate (Maman, Mandelbaum and Zeltyn, 2009). Simulation-based optimization algorithms (Feldman et al., 2008). Comparison between local and global constraint satisfaction. A pot file is a Design Template file, which provides you the “look” of the presentation You apply a pot file by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) or BluePearl Deluxe.pot (white background) and click on Apply. You can switch between black and white background by navigating to that pot file and click on Apply. Another easier way to switch background is by changing color scheme. Opening the Task Pane, select Slide Design – Color Schemes and click on one of the two schemes. All your existing content (including Business Unit or Product Names) will be switched without any modification to color or wording. Start with Blank Presentation, then switch to the desired Design Template Start a new presentation as Blank Presentation You can switch to Blue Onyx Deluxe.pot by opening the Task Pane with View > Task Pane and select Slide Design – Design Templates. Click on the word Browse… at bottom of Task Pane and navigate to where you stored BlueOnyx Deluxe.pot (black background) and click on Apply. Your existing content will take on Blue Onyx’s black background, and previous black text will turn to white. You should add your Business Unit or Product Name by modifying it on the Slide Master You switch to the Slide Master view by View > Master > Slide Master. Click on the Title Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on the Bullet List Page thumbnail icon on the left, and click on the Business Unit or Product Name field to modify it. Click on Close Master View button on the floating Master View Toolbar You can turn on the optional date and footer fields by View > Header and Footer Suggested footer on all pages including Title Page: Presentation Title | Confidential Date and time field can be fixed, or Update automatically. It appears to the right of the footer. Slide number field can be turned on as well. It appears to the left of the footer. 11/15/2018