// Numbas version: finer_feedback_settings
{"name": "MM1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>\\(\\lambda=\\var{lambda}\\) &nbsp;and &nbsp;\\(\\mu=\\var{mu}\\)</p>\n<p></p>\n<p>P(operator is busy) = 1 - P(no calls in the system)</p>\n<p style=\"padding-left: 150px;\">= 1 - \\(P_0\\)</p>\n<p style=\"padding-left: 150px;\">= 1 - \\(\\frac{\\lambda}{\\mu}\\)</p>\n<p style=\"padding-left: 150px;\">&nbsp;= 1 - \\(\\var{p_0}\\)</p>\n<p>Percentage probability = \\(\\simplify{1-{p_0}}*100\\)%</p>\n<p style=\"padding-left: 150px;\">&nbsp; = \\(\\simplify{(1-{p_0})*100}\\)%</p>\n<p></p>\n<p>P(No calls in queue) = P( 0 calls in system OR 1 call in the system)</p>\n<p style=\"padding-left: 150px;\">= \\(P_0+P_1\\)</p>\n<p style=\"padding-left: 150px;\">= \\(\\var{p_0}+\\var{p_1}\\)</p>\n<p style=\"padding-left: 150px;\"></p>\n<p>P(at least n calls in queue) = P(at least n+1 calls in the system)</p>\n<p style=\"padding-left: 180px;\">&nbsp; &nbsp;= 1 - P(less than n+1 calls in the system)</p>\n<p style=\"padding-left: 180px;\">&nbsp; &nbsp;= 1 - {\\(P_0+P_1+....P_n\\)}</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>The mean time a customer will wait for service = \\(W_Q=\\frac{\\lambda}{\\mu(\\mu-\\lambda)}\\)</p>\n<p style=\"padding-left: 150px;\"></p>\n<p></p>", "statement": "<p>A hotel telephone exchange employs one operator to connect&nbsp;incoming and outgoing calls.</p>\n<p>Calls arrive at a mean rate of \\(\\var{lambda}\\) calls per minute. The mean service rate is \\(\\var{mu}\\)&nbsp;calls per minute.</p>", "name": "MM1", "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "ungrouped_variables": ["n", "lambda", "mu", "p_0", "p_1", "p_2", "p_3", "at_least_n"], "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "variables": {"at_least_n": {"description": "", "definition": "if(n=3,1-{p_0}-{p_1}-{p_2}-{p_3},1-{p_0}-{p_1}-{p_2})", "name": "at_least_n", "group": "Ungrouped variables", "templateType": "anything"}, "lambda": {"description": "", "definition": "random(8..18#1)", "name": "lambda", "group": "Ungrouped variables", "templateType": "randrange"}, "mu": {"description": "", "definition": "random(20..32#1)", "name": "mu", "group": "Ungrouped variables", "templateType": "randrange"}, "p_0": {"description": "", "definition": "1-{lambda}/{mu}", "name": "p_0", "group": "Ungrouped variables", "templateType": "anything"}, "p_1": {"description": "", "definition": "{lambda}*{p_0}/{mu}", "name": "p_1", "group": "Ungrouped variables", "templateType": "anything"}, "p_2": {"description": "", "definition": "{lambda}*{p_1}/{mu}", "name": "p_2", "group": "Ungrouped variables", "templateType": "anything"}, "n": {"description": "", "definition": "random(2..3#1)", "name": "n", "group": "Ungrouped variables", "templateType": "randrange"}, "p_3": {"description": "", "definition": "{lambda}*{p_2}/{mu}", "name": "p_3", "group": "Ungrouped variables", "templateType": "anything"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "prompt": "<p>Calculate the percentage probability that the operator is busy.&nbsp;[[0]]</p>\n<p>Calculate the probability that there are no calls queueing for service.&nbsp;[[1]]</p>\n<p>Calculate the probability that there are at least \\(\\var{n}\\) calls in the queue.&nbsp;[[2]]</p>\n<p>Calculate the mean time (minutes) a customer&nbsp;will have to wait for service.&nbsp;[[3]]</p>", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "gaps": [{"showPrecisionHint": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "100*{lambda}/{mu}", "precisionType": "dp", "correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "maxValue": "100*{lambda}/{mu}", "mustBeReducedPC": 0, "precision": 0, "allowFractions": false, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "variableReplacements": [], "scripts": {}, "strictPrecision": false}, {"showPrecisionHint": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{p_0}+{p_1}", "precisionType": "dp", "correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "maxValue": "{p_0}+{p_1}", "mustBeReducedPC": 0, "precision": "2", "allowFractions": false, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "variableReplacements": [], "scripts": {}, "strictPrecision": false}, {"showPrecisionHint": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{at_least_n}", "precisionType": "dp", "correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "maxValue": "{at_least_n}", "mustBeReducedPC": 0, "precision": "3", "allowFractions": false, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "variableReplacements": [], "scripts": {}, "strictPrecision": false}, {"showPrecisionHint": true, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{lambda}/({mu}*({mu}-{lambda}))", "precisionType": "dp", "correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "maxValue": "{lambda}/({mu}*({mu}-{lambda}))", "mustBeReducedPC": 0, "precision": "2", "allowFractions": false, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "variableReplacements": [], "scripts": {}, "strictPrecision": false}]}], "variable_groups": [], "preamble": {"css": "", "js": ""}, "functions": {}, "rulesets": {}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "resources": []}]}], "contributors": [{"name": "Frank Doheny", "profile_url": 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