Study to Evaluate the Effectiveness and Safety of MK-1029 in the Treatment of Persistent Asthma That is Not Controlled With Montelukast (ML) in Adults (MK-1029-011)

The purpose of this study is to evaluate the effect of MK-1029 on lung function in the treatment of adults who have persistent asthma that is uncontrolled with the use of montelukast (ML). Participants will use randomized study drug (either MK-1029 or placebo) for two separate 4-week treatment periods. All participants will also use ML during the treatment periods. The primary hypothesis is that MK-1029 is superior to placebo in change from baseline in forced expiratory volume in one second (FEV1) at the end of the 4-week treatment period.


Eligibility Criteria:
Inclusion Criteria:

* not pregnant or breastfeeding and does not plan to become pregnant for the duration of the study;
* symptoms of persistent asthma for at least one year;
* current use of asthma treatments: 1) "as-needed" inhaled SABAs (albuterol/salbutamol) and no asthma controller for at least 4 weeks prior to Screening Visit OR 2) stable dose of ICS, combination ICS/LABA and/or oral asthma controller(s) for at least 4 weeks prior to Screening Visit and able to tolerate discontinuing all controllers while receiving ML;
* no history of smoking OR no smoking for at least 1 year, with a smoking history of no more than 10 pack-years;
* able to maintain a constant day/night, awake/sleep cycle;
* agrees to not change habitual consumption of beverages or foods containing caffeine throughout the study;
* Body Mass Index (BMI) of 15 kg/m\^2 to 40 kg/m\^2.

Exclusion Criteria:

* history of myocardial infarction, congestive heart failure, or uncontrolled cardiac arrhythmia within 3 months prior to Screening Visit;
* hospitalized or hospitalization within 4 weeks prior to Screening Visit;
* intention of moving or anticipation of missing any study visits;
* any major surgical procedure(s) within 4 weeks prior to Screening Visit;
* participation in a clinical trial involving an investigational drug within 4 weeks prior to Screening Visit;
* current regular use or a recent past abuse (within past 5 years) of alcohol (\>14 drinks/week) or illicit drugs;
* donation of a unit of blood within 2 weeks prior to Screening Visit or intention of donating a unit of blood during the study;
* evidence of another active pulmonary disorder such as bronchiectasis or COPD;
* treatment in an emergency room for asthma within 4 weeks prior to Screening Visit or hospitalization for asthma within 2 months prior to Screening Visit;
* respiratory tract infection which required treatment with antibiotics within 2 months prior to Screening Visit;
* evidence of active sinus disease within 1 week prior to Screening Visit;
* history of a psychiatric disorder, other than stable depression, within 3 months prior to Screening Visit;
* history of human immunodeficiency virus (HIV);
* hypersensitivity or intolerance to inhaled beta-agonists and/or leukotriene inhibitors or any of their ingredients, including lactose and galactose;
* unstable disease of the ophthalmologic, neurological, hepatic, renal, connective tissue, genitourinary, gastrointestinal, cardiovascular or hematologic systems;
* cancer (except for successfully treated basal and squamous cell carcinomas of the skin) or history of cancer within 5 years prior to Screening Visit;
* uncontrolled hypertension.