Queues with system disasters and impatient customers when system is down Uri Yechiali Department of Statistics and Operations Research School of Mathematical Sciences Tel Aviv University, Tel Aviv 69978, Israel (uriy@post.tau.ac.il) Abstract. Consider a system (e.g. a computer farm or a call center) operating as a M/M/c queue, where c = 1, or 1 <c< ∞, or c = ∞. The system as a whole suffers disastrous breakdowns, resulting in the loss of all running and waiting sessions. When the system is down and undergoing a repair process, newly arriving customers become impatient: each individual customer activates a random-duration timer. If the timer expires before the system is repaired, the customer abandons the queue. We analyze this model and derive various quality of service measures: mean sojourn time of a served customer; proportion of customers served; rate of lost customers due to disasters; and rate of abandonments due to impatience. Keywords queues, M/M/1, M/M/c, M/M/∞, failures, disasters, impatience, abandon- ments, sojourn times, quality of service. 1 Introduction Consider a system (e.g. a call center or a computer farm) operating as a M/M/c queue. The system as a whole suffers random failures such that, when a failure occurs, all connections are cut and all existing requests are rejected and lost. The system then goes through a repair process whose duration is random. Meanwhile, while the system is down, the stream of newly arriving requests (customers) continues, but the customers become impatient: each customer ‘activates’ his own ‘timer’ with random duration T , such that, if the system is still down when the timer expires, the customer abandons the system never to return. Our goal is to calculate Quality of Service (QoS) measures: proportion of customers served; rate of customers rejected due to disasters; and rate of abandonments due to impatience when the system is down. 1